Research ArticleDevelopment of High-Resolution Total Variation DiminishingScheme for Linear Hyperbolic Problems

Rabie A Abu Saleem1 and Tomasz Kozlowski2

1Nuclear Engineering Department Jordan University of Science and Technology PO Box 3030 Irbid 22110 Jordan2Department of Nuclear Plasma and Radiological Engineering University of Illinois at Urbana-Champaign 216 Talbot Laboratory104 S Wright Street Urbana IL 61801 USA

Correspondence should be addressed to Rabie A Abu Saleem abusale1illinoisedu

Received 15 March 2015 Revised 7 June 2015 Accepted 21 June 2015

Academic Editor Jim B W Kok

Copyright copy 2015 R A Abu Saleem and T Kozlowski This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A high-resolution total variation diminishing (TVD) stable scheme is derived for scalar hyperbolic problems using the methodof flux limiters The scheme was constructed by combining the 1st-order upwind scheme and the 3rd-order quadratic upstreaminterpolation scheme (QUICK) using new flux limiter function The new flux limiter function was established by imposing severalconditions to ensure the TVD properties of the scheme For temporal discretization the theta method was used and values forthe parameter 120579 were chosen such that the scheme is unconditionally stable Numerical results are presented for one-dimensionalpure advection problems with smooth and discontinuous initial conditions and are compared to those of other known numericalschemes The results show that the proposed numerical method is stable and of higher order than other common schemes

1 Introduction

Development of stable numerical schemes for modelinghyperbolic equations has been a crucial topic for decades ofresearch in the area of computational analysis and simulationThis field of study poses several challenging difficulties Oneof the main difficulties is the formation of different typesof discontinuities in the solution For nonlinear equationsdiscontinuities can appear even for smooth initial condi-tions Different numerical schemes were developed whichexhibit different stability and accuracy problems near thesediscontinuities First-order schemes for example are knownto be stable near discontinuities but they pose an accuracyproblem due to the large numerical dissipations On theother hand higher order linear schemes are less dissipativebut they are known to exhibit nonphysical oscillations neardiscontinuities This will be discussed further in Section 2 ofthis paper A general form of a scalar conservation law in onedimension is given by

120597119902

120597119905+120597119891 (119902)

120597119909= 0 (1)

Equation (1) represents a hyperbolic problem with a char-acteristic speed of 1198911015840(119902) = 120597119891120597119902 This speed representsthe propagation speed of information The solution of thisproblem is constant along the characteristic curves 119909 = 119909

0+

(120597119891120597119902)119905 For nonlinear problems the slopes 120597119891120597119902 (beingdependent on the solution) will not be the same for differentcharacteristic curves which implies either an intersection ordivergence in the 119909-119905 plane The possibility of intersectionmeans multiple values of the solution at some points anddiscontinuities are most likely to appear even for smoothinitial conditions In this context it is a necessity to solvehyperbolic problems using a numerical scheme capable ofhandling discontinuities

Different approaches have been adapted to capture dis-continuous solutions one of which is adding an artificial vis-cosity term to the scheme The addition of artificial viscositycan be achieved either by explicitly adding additional termsor implicitly within the numerical scheme used such as 1st-order schemes (also calledmonotone schemes see Section 2)Other approaches consider adaptive schemes depending onthe smoothness of the solution One way to achieve anadaptive scheme is by considering monotone schemes only

Hindawi Publishing CorporationJournal of Computational EngineeringVolume 2015 Article ID 575380 10 pageshttpdxdoiorg1011552015575380

2 Journal of Computational Engineering

at the regions of discontinuities and higher order schemeselsewhere Examples on this approach are the works doneby Harten [1] Sweby [2] and Kadalbajoo and Kumar [3]Other examples of adaptive schemes are the ENOandWENOschemes where the numerical stencils are adaptive in away that discontinuities are avoided whenever possible [4ndash6] The third approach for handling discontinuous solutionis Godunovrsquos approach where the nonlinearity property ofthe problem is explicitly included in the numerical schemeIn this approach an approximation of the solution is con-structed in the discretized spatial domain and the evolutionof the solution is solved at the interfaces of the spatialcells by solving the so-called Riemann problem exactly orapproximately For more details on the Riemann problem see[7 8] Examples on some numerical methods based on thisapproach can be found in [9 10]

In this paper we present a systematical approach ofderiving an adaptive high-resolution scheme for hyperbolicequations In Section 2 we present numerical notation andconcepts useful for our derivation We proceed with thederivation in Section 3 and present numerical results forsmooth and discontinuous solutions in Section 4 Someconcluding remarks are posed in Section 5

2 Numerical Notations

To allow for discontinuities along a discretized spatial andtemporal domain we need to admit weak solutions for (1) byintegrating over [119909

119894minus12 119909119894+12

] times [119905119899 119905119899+1

] The conservativeform of a numerical scheme in explicit formulation comesdirectly from this process the numerical scheme for (1) canbe written as

119902119899+1119894

= 119902119899

119894minus120582 (F

119899

119894+12 minusF119899

119894minus12) 120579 isin [0 1] (2)

where

119902119899

119894asymp

1Δ119909119894

int

119894+12

119894minus12119902 (119909 119905

119899) 119889119909

F119899

119894+12 asymp1Δ119905

int

119905119899+1

119905119899

119891 (119902 (119909119894+12 119904)) 119889119904

120582 =Δ119905

Δ119909119894

(3)

Other forms of an explicit numerical scheme are the generalform

119902119899+1119894

= 119867 (119902119899

119894minus119896 119902119899

119894minus119896+1 119902119899

119894 119902

119899

119894+119896) (4)

and the incremental form

119902119899+1119894

= 119902119899

119894minus119862119894minus12Δ119902119894minus12 +119863119894+12Δ119902119894+12 (5)

where

Δ119902119894minus12 = 119902

119894minus 119902119894minus1

Δ119902119894+12 = 119902

119894+1 minus 119902119894(6)

A numerical scheme of the general form (4) is called amonotone scheme if the operator 119867 is nondecreasing for allits arguments namely

120597

120597119902119899119896

119867(119902119899

119894minus119897 119902119899

119894minus119897+1 119902119899

119894 119902

119899

119894+119903) ge 0 (7)

for any 119896 in the spatial domain Monotone schemes aredesired because they have the property of not introducingnumerical oscillationswith new extremanear discontinuities

The total variation (TV) is also an important notion thatgives a measure for oscillations due to a numerical schemeFor a given piecewise grid function 119902(119894 119899) that approximatesan exact solution 119902(119909 119905) the total variation is given by [7]

TV (119902119899) =

infin

sum

119894=minusinfin

1003816100381610038161003816119902119899

119894minus 119902119899

119894minus11003816100381610038161003816 (8)

A numerical method is called Total Variation Diminishing(TVD) if the total variation does not increase due to thenumerical operator119867 in (4) mathematically

TV (119902119899+1

) le TV (119902119899) (9)

For a numerical scheme in the incremental form (5) thiscondition is satisfied if [1 11]

119862119894minus12 ge 0

119863119894+12 ge 0

0 le 119862119894minus12 +119863119894+12 le 1

(10)

On the relation between monotonicity and total variationHarten suggests that amonotone scheme has a nonincreasingTV and a scheme with nonincreasing TV preserves mono-tonicity [1]

Godunov proved that when numerical schemes are usedfor solving the advection equation monotonicity cannotbe achieved with linear schemes of order higher than oneThis is referred to as Godunovrsquos order barrier theorem [12]This means that high-order linear schemes such as Lax-Wendroff [13] and the quadratic upstream interpolationscheme (QUICK) [14] cannot be TVD and using suchschemes will introduce nonphysical oscillations wheneverthere is a discontinuity in the solution On the other hand 1st-order schemes are TVD but they exhibit excessive dissipationand tend to drastically smear the solution which in turnmasks physical discontinuities and interfaces To overcomethe drawbacks of the two families of schemes we seek acombination of the two in the form of a nonlinear high-ordermonotone scheme This is achieved by imposing additionalconstraints to ensure the TVD properties of the scheme Inthe next section we present our approach for deriving suchschemes

3 High-Resolution Scheme

In this sectionwepresent a systematical approach for derivinga high-resolution TVD scheme based on the concept of

Journal of Computational Engineering 3

flux limiters This is similar to the work of Sweby [2] andKadalbajoo and Kumar [3] The main idea is to combine theTVD 1st-order upwind scheme and the high-order accurateQUICK schemeby using the first one near discontinuities andthe second one in smooth regions The numerical scheme in(2) is used with a numerical fluxF

1plusmn12= FTVD119894plusmn12

where

FTVD119894plusmn12 = F

1st119894plusmn12 +120593119894plusmn12 [F

QUICK119894plusmn12 minusF

1st119894plusmn12] (11)

F1st119894plusmn12

is the numerical flux associated with a 1st-orderupwind scheme and F

QUICK119894plusmn12

is the numerical flux associ-ated with 3rd-order QUICK scheme Both numerical fluxesdepend on the direction of characteristic speeds and conse-quently obey the hyperbolicity property of the problem

The limiter function 120593119894plusmn12

depends on the smoothness ofthe solution in the computational domain and is chosen suchthat the scheme in (2) is TVD To find the TVD region of thescheme we consider the explicit formulation (120579 = 0) and weapply it to the linear scalar equation for simplicity of analysisnamely

120597119902

120597119905+ 119906

120597119902

120597119909= 0 (12)

For this equation the numerical flux due to the 1st-oderupwind scheme is defined as

119902119891

119894+12 =

119902119894

119906 gt 0

119902119894+1 119906 lt 0

(13)

and for the 3rd-order QUICK scheme

119902119876

119894+12 =

18(6119902119894+ 3119902119894+1 minus 119902

119894minus1) 119906 gt 0

18(6119902119894+1 + 3119902

119894minus 119902119894+2) 119906 lt 0

(14)

If we substitute the numerical fluxes (for the case 119906 gt 0) into(11) and (2) and use 120579 = 0 we obtain the following scheme

119902119899+1119894

= 119902119899

119894minus 119888 [(119902

119894minus 119902119894minus1)

+ 120593119894+12 [

18(6119902119894+ 3119902119894+1 minus 119902119894minus1) minus 119902119894]

minus120593119894minus12 [

18(6119902119894minus1 + 3119902119894 minus 119902119894minus2) minus 119902119894minus1]]

(15)

where 119888 = (119906Δ119905Δ119909) is the Courant number With additionalalgebraic manipulation this can be written as

119902119899+1119894

= 119902119899

119894

minus 119888 [1+120593119894+12

8[3119903119894

+ 1]minus120593119894minus12

8[3+ 119903119894minus1]]

sdot Δ119902119894minus12

(16)

2

18

16

14

12

1

08

06

04

02

00 05 1 15 2 25 3 35 4 45 5

Smoothness parameter (r)

Lim

iter f

unct

ion

8r

3 + r

8

3 + r

120593 =4(2 + r)

3(3 + r)

120593 = 1

Figure 1 Schematic of the TVD region

where 119903119894= Δ119902119894minus12

Δ119902119894+12

is referred to as the smoothnessparameter This is a numerical scheme written in an incre-mental form (5) with the following parameters

119862119894minus12

= (119888 [1+120593119894+12

8[3119903119894

+ 1]minus120593119894minus12

8[3+ 119903119894minus1]])

119863119894+12 = 0

(17)

Implementing the conditions for the scheme to be TVD (10)we obtain the following inequality

minus 8 le 120593119894+12 [

3119903119894

+ 1]minus120593119894minus12 [3+ 119903119894minus1] le 8(1 minus 119888

119888) (18)

At this point we impose another restriction to our scheme byrequiring 0 le 119888 le 12 This yields the following

100381610038161003816100381610038161003816100381610038161003816

120593119894+12 (3 + 119903

119894)

119903119894

minus120593119894minus12 (3+ 119903119894minus1)

100381610038161003816100381610038161003816100381610038161003816

le 8 (19)

For our scheme to be TVD at extreme points we must have120593(119903) = 0 for 119903 lt 0 This condition along with (19) is satisfiedfor the bounds

0 le (120593119894+12 (3 + 119903

119894)

119903119894

120593119894minus12 (3+ 119903119894minus1)) le 8 (20)

Equation (20) defines the TVD region shaded in Figure 1This region is bounded by the function 120593(119903) = min[8119903(3 +119903) 8(3 + 119903)] The same TVD region can be obtained for thecase of negative advection velocity For the numerical schemeto be TVD the limiter function used has to lie in the TVDshaded area of Figure 1

4 Journal of Computational Engineering

It is obvious that maximizing the flux limiter increasesthe antidiffusivity of the scheme This means that a fluxlimiter function corresponding to the upper boundary ofthe TVD region yields the least diffusive scheme possibleNevertheless choosing such a limiter does not guarantee thehighest possible order of accuracy To ensure high-order ofaccuracy we require one additional condition the limiterfunction should be chosen such that the scheme in (16) is 3rd-order accurate whenever possible To impose this constraintwe investigated another numerical scheme and the 3rd-orderupwind scheme For this scheme the numerical fluxes aredefined as

119902119880

119894+12 =

16(5119902119894minus 119902119894minus1 + 2119902

119894+1) 119906 gt 0

16(5119902119894+1 minus 119902

119894+2 + 2119902119894) 119906 lt 0

(21)

The discretization for the advection equation (12) due to thisscheme (for 119906 gt 0) is given by

119902119899+1119894

= 119902119899

119894

minusΔ119905

Δ119909119906 [

5119902119894minus 119902119894minus1 + 2119902

119894+16

minus5119902119894minus1 minus 119902

119894minus2 + 2119902119894

6]

(22)

With algebraic manipulation it can be shown that thisscheme is equivalent to the scheme in (16) when the fluxlimiter function is chosen to be 120593 = 4(2 + 119903)3(3 + 119903) Hencewe define the darker shaded region in Figure 1 as the desired3rd-order TVD region for our scheme

To guarantee all the conditions we imposed are satisfiedwe derive a family of flux limiters lying in the 3rd-order TVDregion (darker shaded region) whenever possible Any fluxfunction in that region can be expressed as an arithmeticaverage of two limiter functions 120593 = 1 that corresponds tothe QUICK scheme and 120593 = 4(2 + 119903)3(3 + 119903) for the 3rd-order upwind scheme namely

120593lowast= 1+ 120575 (119903 minus 1)

3 (3 + 119903) 0 lt 120575 lt 1 (23)

Intersection points 119903lowast1and 119903lowast2are found to be 119903lowast

1= (9minus120575)(21minus

120575) and 119903lowast

2= (120575 + 15)(120575 + 3) (see Figure 2) We can see

that for any chosen value of 120575 the limiter function satisfiesour condition for smooth regions 120593lowast(1) = 1 The arithmeticaverage and the intersection points are shown in Figure 2

Based on the above we propose the following family offlux limiters characterized by the parameter 120575

120593 (119903) =

0 119903 lt 0

81199033 + 119903

0 lt 119903 lt9 minus 120575

21 minus 120575

1 + 120575 (119903 minus 1)3 (3 + 119903)

9 minus 120575

21 minus 120575lt 119903 lt

120575 + 15120575 + 3

83 + 119903

120575 + 15120575 + 3

lt 119903

(24)

120593 =4(2 + r)

3(3 + r)

2

18

16

14

12

1

08

06

04

02

00 15 21 25 3 35 4 5

Smoothness parameter (r)

Lim

iter f

unct

ion

120593 = 1

rlowast2

120593lowast = 1 +120575(r minus 1)

3(3 + r)

rlowast1

Figure 2 Arithmetically averaged limiter function

Equation (24) shows that the proposed limiter function iscontinuous and bounded for any value of the smoothnessparameter 119903 It also satisfies the condition for smooth regions120593(1) = 1 The flux limiter functions in (24) lie in the TVDregion for any choice of 120575 hence the scheme is TVD

In this paper the theta method (also called the gen-eralized Crank-Nicolson method) is used by consideringa weighted average of explicit and implicit terms in thenumerical scheme of (2) The scheme can be written as

119902119899+1119894

= 119902119899

119894minus120582 [120579 (F

119899+1119894+12 minusF

119899+1119894minus12)

+ (1minus 120579) (F119899119894+12 minusF

119899

119894minus12)] 120579 isin [0 1] (25)

Equation (25) represents a family of schemes characterizedby 120579 We obtain the explicit scheme for 120579 = 0 the implicitscheme for 120579 = 1 and the Crank-Nicolson scheme for 120579 =

12The thetamethod is unconditionally stable for any choiceof 120579 isin [12 1] [15 16] Spatially the accuracy of the schemeis determined by the way of calculating the numerical fluxesF119899119894plusmn12

4 Numerical Results and Discussion

In this section we present two sets of numerical resultsone for the case of smooth solutions and the other fordiscontinuous solutions Both sets of results are shown forthe case of pure advection with positive velocity Results fromthe proposed scheme are also compared to those from othernumerical schemes

41 Smooth Initial Conditions For this set of results weconsider (12) with a smooth initial condition given by

1199020 (119909) = 05+ 03 sin (2120587119909) (26)

Journal of Computational Engineering 5

0

05

1

0 05 1

0

05

1

005 1

0

05

1

0 05 1

0

05

1

0 05

1

1st-order upwind Lax-Wendroff

2nd-order upwind Minbee limiter

q q

q q

x

x

0

05

1

0 05 1

q

x

0

05

1

0 05

x

1

q

0

05

1

0 05

x

1

q

0

05

1

0 05 1

q

x

x

x

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 3 Numerical results for smooth solutions with 120579 = 1

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain Total timeof the simulation was set to 1 sec so that the solution at theend of the simulation is the same as the initial condition

Figure 3 shows a comparison of the new scheme withother classical schemes and existing high-resolution schemesfor the case of implicit time discretization (120579 = 1) Compari-son was made for a grid size of 005m and a Courant numberof 02

Excessive dissipation of the 1st-order upwind schemecan be observed and the solution is smeared out due tothe inherent numerical viscosity of the scheme For the twohigher order linear schemes (2nd-order upwind scheme and

Lax-Wendroff scheme) dissipative errors were minimizedbut the signal in the numerical solution was either leadingor lagging due to dispersion errors This dispersive nature ofthe error can be observed by studying the modified equationfor the corresponding schemes For example the modifiedequation corresponding to the 2nd-order upwind scheme is

120597119902

120597119905+ 119906

120597119902

120597119909= 119906

2Δ119905 (120579 minus

12)1205972119902

1205971199092

+(119906Δ119909

2

3minus1199063Δ119905

2

2(120579minus

13))

1205973119902

1205971199093

(27)

6 Journal of Computational Engineering

05

1

00

05 1

1st-order upwind

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

Lax-Wendroff

2nd-order upwind Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 4 Numerical results for smooth solutions with 120579 = 12

It can be seen from (27) that the 2nd-order upwind schemeposes two types of errors dissipation errors due to temporaldiscretization and dispersive errors due to both temporal andspatial discretization The latter type of error causes eitherleading or lagging of the numerical solution as shown inFigure 3

On the other hand more satisfactory results can beobserved when implementing high- resolution schemes withdifferent flux limitersThe new scheme with 120575 = 1 resulted ina better numerical solution as compared to other schemes

Figure 4 shows a comparison of the same set of schemesfor the case of Crank-Nicolson time discretization (120579 = 12)By examining (27) it is clear that the dissipative error termscease to exist for the case of 120579 = 12 This explains the

reduction in dissipation for the high-order linear schemes(2nd-order upwind and Lax-Wendroff) On the other handour new scheme continues to produce the best results ascompared to other schemes

The exact solution was used to determine the accuracy ofdifferent numerical schemes Accuracy was assessed on thebasis of the 119871

1norm defined as

1198711 =1119873

119873

sum

119894=1

1003816100381610038161003816119902119873 (119909119894) minus 119902119864 (119909119894)1003816100381610038161003816 (28)

where 119873 is the number of cells and 119902119873(119909119894) is the numerical

solution at the 119894th cell Convergence rate is represented bya log-log plot of the norm of the error versus mesh size

Journal of Computational Engineering 7

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(

erro

r)

1st-order upwindQUICK schemeMinbee limiter

Superbee limiterKumar limiter

log(dx)

Suggested limiter (delta = 1)

Figure 5 Spatial convergence rates for smooth solutions

Table 1 Spatial convergence rates of different schemes for smoothsolution

Numerical scheme Convergence rate1st-order upwind 086QUICK scheme 13Minbee limiter 11Superbee 099Kumar limiter 13New limiter with 120575 = 1 13

Figure 5 shows the results of the spatial convergence for thecase 120579 = 12 Convergence rates for several consideredschemes are listed in Table 1 The results show competitiverate of convergence for the new scheme as compared to otherschemes

42 Discontinuous Initial Conditions For this set of resultswe consider (12) with a discontinuous initial condition givenby

1199020 (119909) =

03 for 00 lt 119909 lt 04

08 for 04 lt 119909 lt 06

03 for 06 lt 119909 lt 10

(29)

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain and the timeof the simulation is 1 sec

Table 2 Spatial convergence rates of different schemes for discon-tinuous solution

Numerical scheme Convergence rate1st-order upwind 040Minbee limiter 056Superbee 052Kumar limiter 070Suggested limiter with 120575 = 1 082

Two sets of results are shown implicit time discretization(120579 = 1) is shown in Figure 6 and Crank-Nicolson timediscretization is shown in Figure 7 Comparison is made fora grid size of 00125m and Courant number of 02

Results show significantly more dissipation for the caseof implicit discretization For the 2nd-order upwind schemeand the QUICK scheme dispersion error appears in theformof overshoots and undershoots For theQUICK schemethese oscillations were damped for the case of 120579 = 1

due to dissipation High-resolution schemes including thenew scheme give better results capturing the discontinuityespecially for the case with 120579 = 12 where dissipation isreduced

Figure 8 shows results of the spatial convergence study fordifferent schemes Results of this study are listed in Table 2We can observe a very good rate of convergence for the newscheme compared to the other schemes

5 Concluding Remarks

A new high-resolution stable scheme is derived by imple-menting a hybridization of the monotone 1st-order upwindscheme and the quadratic upstream interpolation scheme(QUICK) For temporal discretization the generalizedCrank-Nicolson method characterized by the parameter 120579was used The combination of the 1st-order upwind schemeand QUICK scheme was done by means of a flux limitingfunction which depends on the smoothness of the solutionConstraints were imposed to make the resulting schemeTVD and high-order at smooth regions of the solution Forthe one-dimensional linear case results from the proposedscheme for different values of 120579 were compared to classicaland popular high-resolution schemes and the convergencerate for the scheme was investigated for smooth and dis-continuous solutions The proposed scheme was shown toexhibit very good results as compared to other schemes Itis shown that the high-resolution schemes based on 2ndorder Lax-Wendroff discretization exhibit a convergence rateof about 1 for smooth solutions and 05 for discontinuoussolutions The new scheme based on 3rd-order QUICKdiscretization exhibits a convergence rate of about 13 forsmooth solutions and 08 for discontinuous solutions In thecase of discontinuous solutions the new method derived inthis paper shows the best convergence rate of all schemesinvestigated here

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

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2 Journal of Computational Engineering

at the regions of discontinuities and higher order schemeselsewhere Examples on this approach are the works doneby Harten [1] Sweby [2] and Kadalbajoo and Kumar [3]Other examples of adaptive schemes are the ENOandWENOschemes where the numerical stencils are adaptive in away that discontinuities are avoided whenever possible [4ndash6] The third approach for handling discontinuous solutionis Godunovrsquos approach where the nonlinearity property ofthe problem is explicitly included in the numerical schemeIn this approach an approximation of the solution is con-structed in the discretized spatial domain and the evolutionof the solution is solved at the interfaces of the spatialcells by solving the so-called Riemann problem exactly orapproximately For more details on the Riemann problem see[7 8] Examples on some numerical methods based on thisapproach can be found in [9 10]

In this paper we present a systematical approach ofderiving an adaptive high-resolution scheme for hyperbolicequations In Section 2 we present numerical notation andconcepts useful for our derivation We proceed with thederivation in Section 3 and present numerical results forsmooth and discontinuous solutions in Section 4 Someconcluding remarks are posed in Section 5

2 Numerical Notations

To allow for discontinuities along a discretized spatial andtemporal domain we need to admit weak solutions for (1) byintegrating over [119909

119894minus12 119909119894+12

] times [119905119899 119905119899+1

] The conservativeform of a numerical scheme in explicit formulation comesdirectly from this process the numerical scheme for (1) canbe written as

119902119899+1119894

= 119902119899

119894minus120582 (F

119899

119894+12 minusF119899

119894minus12) 120579 isin [0 1] (2)

where

119902119899

119894asymp

1Δ119909119894

int

119894+12

119894minus12119902 (119909 119905

119899) 119889119909

F119899

119894+12 asymp1Δ119905

int

119905119899+1

119905119899

119891 (119902 (119909119894+12 119904)) 119889119904

120582 =Δ119905

Δ119909119894

(3)

Other forms of an explicit numerical scheme are the generalform

119902119899+1119894

= 119867 (119902119899

119894minus119896 119902119899

119894minus119896+1 119902119899

119894 119902

119899

119894+119896) (4)

and the incremental form

119902119899+1119894

= 119902119899

119894minus119862119894minus12Δ119902119894minus12 +119863119894+12Δ119902119894+12 (5)

where

Δ119902119894minus12 = 119902

119894minus 119902119894minus1

Δ119902119894+12 = 119902

119894+1 minus 119902119894(6)

A numerical scheme of the general form (4) is called amonotone scheme if the operator 119867 is nondecreasing for allits arguments namely

120597

120597119902119899119896

119867(119902119899

119894minus119897 119902119899

119894minus119897+1 119902119899

119894 119902

119899

119894+119903) ge 0 (7)

for any 119896 in the spatial domain Monotone schemes aredesired because they have the property of not introducingnumerical oscillationswith new extremanear discontinuities

The total variation (TV) is also an important notion thatgives a measure for oscillations due to a numerical schemeFor a given piecewise grid function 119902(119894 119899) that approximatesan exact solution 119902(119909 119905) the total variation is given by [7]

TV (119902119899) =

infin

sum

119894=minusinfin

1003816100381610038161003816119902119899

119894minus 119902119899

119894minus11003816100381610038161003816 (8)

A numerical method is called Total Variation Diminishing(TVD) if the total variation does not increase due to thenumerical operator119867 in (4) mathematically

TV (119902119899+1

) le TV (119902119899) (9)

For a numerical scheme in the incremental form (5) thiscondition is satisfied if [1 11]

119862119894minus12 ge 0

119863119894+12 ge 0

0 le 119862119894minus12 +119863119894+12 le 1

(10)

On the relation between monotonicity and total variationHarten suggests that amonotone scheme has a nonincreasingTV and a scheme with nonincreasing TV preserves mono-tonicity [1]

Godunov proved that when numerical schemes are usedfor solving the advection equation monotonicity cannotbe achieved with linear schemes of order higher than oneThis is referred to as Godunovrsquos order barrier theorem [12]This means that high-order linear schemes such as Lax-Wendroff [13] and the quadratic upstream interpolationscheme (QUICK) [14] cannot be TVD and using suchschemes will introduce nonphysical oscillations wheneverthere is a discontinuity in the solution On the other hand 1st-order schemes are TVD but they exhibit excessive dissipationand tend to drastically smear the solution which in turnmasks physical discontinuities and interfaces To overcomethe drawbacks of the two families of schemes we seek acombination of the two in the form of a nonlinear high-ordermonotone scheme This is achieved by imposing additionalconstraints to ensure the TVD properties of the scheme Inthe next section we present our approach for deriving suchschemes

3 High-Resolution Scheme

In this sectionwepresent a systematical approach for derivinga high-resolution TVD scheme based on the concept of

Journal of Computational Engineering 3

flux limiters This is similar to the work of Sweby [2] andKadalbajoo and Kumar [3] The main idea is to combine theTVD 1st-order upwind scheme and the high-order accurateQUICK schemeby using the first one near discontinuities andthe second one in smooth regions The numerical scheme in(2) is used with a numerical fluxF

1plusmn12= FTVD119894plusmn12

where

FTVD119894plusmn12 = F

1st119894plusmn12 +120593119894plusmn12 [F

QUICK119894plusmn12 minusF

1st119894plusmn12] (11)

F1st119894plusmn12

is the numerical flux associated with a 1st-orderupwind scheme and F

QUICK119894plusmn12

is the numerical flux associ-ated with 3rd-order QUICK scheme Both numerical fluxesdepend on the direction of characteristic speeds and conse-quently obey the hyperbolicity property of the problem

The limiter function 120593119894plusmn12

depends on the smoothness ofthe solution in the computational domain and is chosen suchthat the scheme in (2) is TVD To find the TVD region of thescheme we consider the explicit formulation (120579 = 0) and weapply it to the linear scalar equation for simplicity of analysisnamely

120597119902

120597119905+ 119906

120597119902

120597119909= 0 (12)

For this equation the numerical flux due to the 1st-oderupwind scheme is defined as

119902119891

119894+12 =

119902119894

119906 gt 0

119902119894+1 119906 lt 0

(13)

and for the 3rd-order QUICK scheme

119902119876

119894+12 =

18(6119902119894+ 3119902119894+1 minus 119902

119894minus1) 119906 gt 0

18(6119902119894+1 + 3119902

119894minus 119902119894+2) 119906 lt 0

(14)

If we substitute the numerical fluxes (for the case 119906 gt 0) into(11) and (2) and use 120579 = 0 we obtain the following scheme

119902119899+1119894

= 119902119899

119894minus 119888 [(119902

119894minus 119902119894minus1)

+ 120593119894+12 [

18(6119902119894+ 3119902119894+1 minus 119902119894minus1) minus 119902119894]

minus120593119894minus12 [

18(6119902119894minus1 + 3119902119894 minus 119902119894minus2) minus 119902119894minus1]]

(15)

where 119888 = (119906Δ119905Δ119909) is the Courant number With additionalalgebraic manipulation this can be written as

119902119899+1119894

= 119902119899

119894

minus 119888 [1+120593119894+12

8[3119903119894

+ 1]minus120593119894minus12

8[3+ 119903119894minus1]]

sdot Δ119902119894minus12

(16)

2

18

16

14

12

1

08

06

04

02

00 05 1 15 2 25 3 35 4 45 5

Smoothness parameter (r)

Lim

iter f

unct

ion

8r

3 + r

8

3 + r

120593 =4(2 + r)

3(3 + r)

120593 = 1

Figure 1 Schematic of the TVD region

where 119903119894= Δ119902119894minus12

Δ119902119894+12

is referred to as the smoothnessparameter This is a numerical scheme written in an incre-mental form (5) with the following parameters

119862119894minus12

= (119888 [1+120593119894+12

8[3119903119894

+ 1]minus120593119894minus12

8[3+ 119903119894minus1]])

119863119894+12 = 0

(17)

Implementing the conditions for the scheme to be TVD (10)we obtain the following inequality

minus 8 le 120593119894+12 [

3119903119894

+ 1]minus120593119894minus12 [3+ 119903119894minus1] le 8(1 minus 119888

119888) (18)

At this point we impose another restriction to our scheme byrequiring 0 le 119888 le 12 This yields the following

100381610038161003816100381610038161003816100381610038161003816

120593119894+12 (3 + 119903

119894)

119903119894

minus120593119894minus12 (3+ 119903119894minus1)

100381610038161003816100381610038161003816100381610038161003816

le 8 (19)

For our scheme to be TVD at extreme points we must have120593(119903) = 0 for 119903 lt 0 This condition along with (19) is satisfiedfor the bounds

0 le (120593119894+12 (3 + 119903

119894)

119903119894

120593119894minus12 (3+ 119903119894minus1)) le 8 (20)

Equation (20) defines the TVD region shaded in Figure 1This region is bounded by the function 120593(119903) = min[8119903(3 +119903) 8(3 + 119903)] The same TVD region can be obtained for thecase of negative advection velocity For the numerical schemeto be TVD the limiter function used has to lie in the TVDshaded area of Figure 1

4 Journal of Computational Engineering

It is obvious that maximizing the flux limiter increasesthe antidiffusivity of the scheme This means that a fluxlimiter function corresponding to the upper boundary ofthe TVD region yields the least diffusive scheme possibleNevertheless choosing such a limiter does not guarantee thehighest possible order of accuracy To ensure high-order ofaccuracy we require one additional condition the limiterfunction should be chosen such that the scheme in (16) is 3rd-order accurate whenever possible To impose this constraintwe investigated another numerical scheme and the 3rd-orderupwind scheme For this scheme the numerical fluxes aredefined as

119902119880

119894+12 =

16(5119902119894minus 119902119894minus1 + 2119902

119894+1) 119906 gt 0

16(5119902119894+1 minus 119902

119894+2 + 2119902119894) 119906 lt 0

(21)

The discretization for the advection equation (12) due to thisscheme (for 119906 gt 0) is given by

119902119899+1119894

= 119902119899

119894

minusΔ119905

Δ119909119906 [

5119902119894minus 119902119894minus1 + 2119902

119894+16

minus5119902119894minus1 minus 119902

119894minus2 + 2119902119894

6]

(22)

With algebraic manipulation it can be shown that thisscheme is equivalent to the scheme in (16) when the fluxlimiter function is chosen to be 120593 = 4(2 + 119903)3(3 + 119903) Hencewe define the darker shaded region in Figure 1 as the desired3rd-order TVD region for our scheme

To guarantee all the conditions we imposed are satisfiedwe derive a family of flux limiters lying in the 3rd-order TVDregion (darker shaded region) whenever possible Any fluxfunction in that region can be expressed as an arithmeticaverage of two limiter functions 120593 = 1 that corresponds tothe QUICK scheme and 120593 = 4(2 + 119903)3(3 + 119903) for the 3rd-order upwind scheme namely

120593lowast= 1+ 120575 (119903 minus 1)

3 (3 + 119903) 0 lt 120575 lt 1 (23)

Intersection points 119903lowast1and 119903lowast2are found to be 119903lowast

1= (9minus120575)(21minus

120575) and 119903lowast

2= (120575 + 15)(120575 + 3) (see Figure 2) We can see

that for any chosen value of 120575 the limiter function satisfiesour condition for smooth regions 120593lowast(1) = 1 The arithmeticaverage and the intersection points are shown in Figure 2

Based on the above we propose the following family offlux limiters characterized by the parameter 120575

120593 (119903) =

0 119903 lt 0

81199033 + 119903

0 lt 119903 lt9 minus 120575

21 minus 120575

1 + 120575 (119903 minus 1)3 (3 + 119903)

9 minus 120575

21 minus 120575lt 119903 lt

120575 + 15120575 + 3

83 + 119903

120575 + 15120575 + 3

lt 119903

(24)

120593 =4(2 + r)

3(3 + r)

2

18

16

14

12

1

08

06

04

02

00 15 21 25 3 35 4 5

Smoothness parameter (r)

Lim

iter f

unct

ion

120593 = 1

rlowast2

120593lowast = 1 +120575(r minus 1)

3(3 + r)

rlowast1

Figure 2 Arithmetically averaged limiter function

Equation (24) shows that the proposed limiter function iscontinuous and bounded for any value of the smoothnessparameter 119903 It also satisfies the condition for smooth regions120593(1) = 1 The flux limiter functions in (24) lie in the TVDregion for any choice of 120575 hence the scheme is TVD

In this paper the theta method (also called the gen-eralized Crank-Nicolson method) is used by consideringa weighted average of explicit and implicit terms in thenumerical scheme of (2) The scheme can be written as

119902119899+1119894

= 119902119899

119894minus120582 [120579 (F

119899+1119894+12 minusF

119899+1119894minus12)

+ (1minus 120579) (F119899119894+12 minusF

119899

119894minus12)] 120579 isin [0 1] (25)

Equation (25) represents a family of schemes characterizedby 120579 We obtain the explicit scheme for 120579 = 0 the implicitscheme for 120579 = 1 and the Crank-Nicolson scheme for 120579 =

12The thetamethod is unconditionally stable for any choiceof 120579 isin [12 1] [15 16] Spatially the accuracy of the schemeis determined by the way of calculating the numerical fluxesF119899119894plusmn12

4 Numerical Results and Discussion

In this section we present two sets of numerical resultsone for the case of smooth solutions and the other fordiscontinuous solutions Both sets of results are shown forthe case of pure advection with positive velocity Results fromthe proposed scheme are also compared to those from othernumerical schemes

41 Smooth Initial Conditions For this set of results weconsider (12) with a smooth initial condition given by

1199020 (119909) = 05+ 03 sin (2120587119909) (26)

Journal of Computational Engineering 5

0

05

1

0 05 1

0

05

1

005 1

0

05

1

0 05 1

0

05

1

0 05

1

1st-order upwind Lax-Wendroff

2nd-order upwind Minbee limiter

q q

q q

x

x

0

05

1

0 05 1

q

x

0

05

1

0 05

x

1

q

0

05

1

0 05

x

1

q

0

05

1

0 05 1

q

x

x

x

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 3 Numerical results for smooth solutions with 120579 = 1

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain Total timeof the simulation was set to 1 sec so that the solution at theend of the simulation is the same as the initial condition

Figure 3 shows a comparison of the new scheme withother classical schemes and existing high-resolution schemesfor the case of implicit time discretization (120579 = 1) Compari-son was made for a grid size of 005m and a Courant numberof 02

Excessive dissipation of the 1st-order upwind schemecan be observed and the solution is smeared out due tothe inherent numerical viscosity of the scheme For the twohigher order linear schemes (2nd-order upwind scheme and

Lax-Wendroff scheme) dissipative errors were minimizedbut the signal in the numerical solution was either leadingor lagging due to dispersion errors This dispersive nature ofthe error can be observed by studying the modified equationfor the corresponding schemes For example the modifiedequation corresponding to the 2nd-order upwind scheme is

120597119902

120597119905+ 119906

120597119902

120597119909= 119906

2Δ119905 (120579 minus

12)1205972119902

1205971199092

+(119906Δ119909

2

3minus1199063Δ119905

2

2(120579minus

13))

1205973119902

1205971199093

(27)

6 Journal of Computational Engineering

05

1

00

05 1

1st-order upwind

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

Lax-Wendroff

2nd-order upwind Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 4 Numerical results for smooth solutions with 120579 = 12

It can be seen from (27) that the 2nd-order upwind schemeposes two types of errors dissipation errors due to temporaldiscretization and dispersive errors due to both temporal andspatial discretization The latter type of error causes eitherleading or lagging of the numerical solution as shown inFigure 3

On the other hand more satisfactory results can beobserved when implementing high- resolution schemes withdifferent flux limitersThe new scheme with 120575 = 1 resulted ina better numerical solution as compared to other schemes

Figure 4 shows a comparison of the same set of schemesfor the case of Crank-Nicolson time discretization (120579 = 12)By examining (27) it is clear that the dissipative error termscease to exist for the case of 120579 = 12 This explains the

reduction in dissipation for the high-order linear schemes(2nd-order upwind and Lax-Wendroff) On the other handour new scheme continues to produce the best results ascompared to other schemes

The exact solution was used to determine the accuracy ofdifferent numerical schemes Accuracy was assessed on thebasis of the 119871

1norm defined as

1198711 =1119873

119873

sum

119894=1

1003816100381610038161003816119902119873 (119909119894) minus 119902119864 (119909119894)1003816100381610038161003816 (28)

where 119873 is the number of cells and 119902119873(119909119894) is the numerical

solution at the 119894th cell Convergence rate is represented bya log-log plot of the norm of the error versus mesh size

Journal of Computational Engineering 7

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(

erro

r)

1st-order upwindQUICK schemeMinbee limiter

Superbee limiterKumar limiter

log(dx)

Suggested limiter (delta = 1)

Figure 5 Spatial convergence rates for smooth solutions

Table 1 Spatial convergence rates of different schemes for smoothsolution

Numerical scheme Convergence rate1st-order upwind 086QUICK scheme 13Minbee limiter 11Superbee 099Kumar limiter 13New limiter with 120575 = 1 13

Figure 5 shows the results of the spatial convergence for thecase 120579 = 12 Convergence rates for several consideredschemes are listed in Table 1 The results show competitiverate of convergence for the new scheme as compared to otherschemes

42 Discontinuous Initial Conditions For this set of resultswe consider (12) with a discontinuous initial condition givenby

1199020 (119909) =

03 for 00 lt 119909 lt 04

08 for 04 lt 119909 lt 06

03 for 06 lt 119909 lt 10

(29)

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain and the timeof the simulation is 1 sec

Table 2 Spatial convergence rates of different schemes for discon-tinuous solution

Numerical scheme Convergence rate1st-order upwind 040Minbee limiter 056Superbee 052Kumar limiter 070Suggested limiter with 120575 = 1 082

Two sets of results are shown implicit time discretization(120579 = 1) is shown in Figure 6 and Crank-Nicolson timediscretization is shown in Figure 7 Comparison is made fora grid size of 00125m and Courant number of 02

Results show significantly more dissipation for the caseof implicit discretization For the 2nd-order upwind schemeand the QUICK scheme dispersion error appears in theformof overshoots and undershoots For theQUICK schemethese oscillations were damped for the case of 120579 = 1

due to dissipation High-resolution schemes including thenew scheme give better results capturing the discontinuityespecially for the case with 120579 = 12 where dissipation isreduced

Figure 8 shows results of the spatial convergence study fordifferent schemes Results of this study are listed in Table 2We can observe a very good rate of convergence for the newscheme compared to the other schemes

5 Concluding Remarks

A new high-resolution stable scheme is derived by imple-menting a hybridization of the monotone 1st-order upwindscheme and the quadratic upstream interpolation scheme(QUICK) For temporal discretization the generalizedCrank-Nicolson method characterized by the parameter 120579was used The combination of the 1st-order upwind schemeand QUICK scheme was done by means of a flux limitingfunction which depends on the smoothness of the solutionConstraints were imposed to make the resulting schemeTVD and high-order at smooth regions of the solution Forthe one-dimensional linear case results from the proposedscheme for different values of 120579 were compared to classicaland popular high-resolution schemes and the convergencerate for the scheme was investigated for smooth and dis-continuous solutions The proposed scheme was shown toexhibit very good results as compared to other schemes Itis shown that the high-resolution schemes based on 2ndorder Lax-Wendroff discretization exhibit a convergence rateof about 1 for smooth solutions and 05 for discontinuoussolutions The new scheme based on 3rd-order QUICKdiscretization exhibits a convergence rate of about 13 forsmooth solutions and 08 for discontinuous solutions In thecase of discontinuous solutions the new method derived inthis paper shows the best convergence rate of all schemesinvestigated here

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

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Journal of Computational Engineering 3

flux limiters This is similar to the work of Sweby [2] andKadalbajoo and Kumar [3] The main idea is to combine theTVD 1st-order upwind scheme and the high-order accurateQUICK schemeby using the first one near discontinuities andthe second one in smooth regions The numerical scheme in(2) is used with a numerical fluxF

1plusmn12= FTVD119894plusmn12

where

FTVD119894plusmn12 = F

1st119894plusmn12 +120593119894plusmn12 [F

QUICK119894plusmn12 minusF

1st119894plusmn12] (11)

F1st119894plusmn12

is the numerical flux associated with a 1st-orderupwind scheme and F

QUICK119894plusmn12

is the numerical flux associ-ated with 3rd-order QUICK scheme Both numerical fluxesdepend on the direction of characteristic speeds and conse-quently obey the hyperbolicity property of the problem

The limiter function 120593119894plusmn12

depends on the smoothness ofthe solution in the computational domain and is chosen suchthat the scheme in (2) is TVD To find the TVD region of thescheme we consider the explicit formulation (120579 = 0) and weapply it to the linear scalar equation for simplicity of analysisnamely

120597119902

120597119905+ 119906

120597119902

120597119909= 0 (12)

For this equation the numerical flux due to the 1st-oderupwind scheme is defined as

119902119891

119894+12 =

119902119894

119906 gt 0

119902119894+1 119906 lt 0

(13)

and for the 3rd-order QUICK scheme

119902119876

119894+12 =

18(6119902119894+ 3119902119894+1 minus 119902

119894minus1) 119906 gt 0

18(6119902119894+1 + 3119902

119894minus 119902119894+2) 119906 lt 0

(14)

If we substitute the numerical fluxes (for the case 119906 gt 0) into(11) and (2) and use 120579 = 0 we obtain the following scheme

119902119899+1119894

= 119902119899

119894minus 119888 [(119902

119894minus 119902119894minus1)

+ 120593119894+12 [

18(6119902119894+ 3119902119894+1 minus 119902119894minus1) minus 119902119894]

minus120593119894minus12 [

18(6119902119894minus1 + 3119902119894 minus 119902119894minus2) minus 119902119894minus1]]

(15)

where 119888 = (119906Δ119905Δ119909) is the Courant number With additionalalgebraic manipulation this can be written as

119902119899+1119894

= 119902119899

119894

minus 119888 [1+120593119894+12

8[3119903119894

+ 1]minus120593119894minus12

8[3+ 119903119894minus1]]

sdot Δ119902119894minus12

(16)

2

18

16

14

12

1

08

06

04

02

00 05 1 15 2 25 3 35 4 45 5

Smoothness parameter (r)

Lim

iter f

unct

ion

8r

3 + r

8

3 + r

120593 =4(2 + r)

3(3 + r)

120593 = 1

Figure 1 Schematic of the TVD region

where 119903119894= Δ119902119894minus12

Δ119902119894+12

is referred to as the smoothnessparameter This is a numerical scheme written in an incre-mental form (5) with the following parameters

119862119894minus12

= (119888 [1+120593119894+12

8[3119903119894

+ 1]minus120593119894minus12

8[3+ 119903119894minus1]])

119863119894+12 = 0

(17)

Implementing the conditions for the scheme to be TVD (10)we obtain the following inequality

minus 8 le 120593119894+12 [

3119903119894

+ 1]minus120593119894minus12 [3+ 119903119894minus1] le 8(1 minus 119888

119888) (18)

At this point we impose another restriction to our scheme byrequiring 0 le 119888 le 12 This yields the following

100381610038161003816100381610038161003816100381610038161003816

120593119894+12 (3 + 119903

119894)

119903119894

minus120593119894minus12 (3+ 119903119894minus1)

100381610038161003816100381610038161003816100381610038161003816

le 8 (19)

For our scheme to be TVD at extreme points we must have120593(119903) = 0 for 119903 lt 0 This condition along with (19) is satisfiedfor the bounds

0 le (120593119894+12 (3 + 119903

119894)

119903119894

120593119894minus12 (3+ 119903119894minus1)) le 8 (20)

Equation (20) defines the TVD region shaded in Figure 1This region is bounded by the function 120593(119903) = min[8119903(3 +119903) 8(3 + 119903)] The same TVD region can be obtained for thecase of negative advection velocity For the numerical schemeto be TVD the limiter function used has to lie in the TVDshaded area of Figure 1

4 Journal of Computational Engineering

It is obvious that maximizing the flux limiter increasesthe antidiffusivity of the scheme This means that a fluxlimiter function corresponding to the upper boundary ofthe TVD region yields the least diffusive scheme possibleNevertheless choosing such a limiter does not guarantee thehighest possible order of accuracy To ensure high-order ofaccuracy we require one additional condition the limiterfunction should be chosen such that the scheme in (16) is 3rd-order accurate whenever possible To impose this constraintwe investigated another numerical scheme and the 3rd-orderupwind scheme For this scheme the numerical fluxes aredefined as

119902119880

119894+12 =

16(5119902119894minus 119902119894minus1 + 2119902

119894+1) 119906 gt 0

16(5119902119894+1 minus 119902

119894+2 + 2119902119894) 119906 lt 0

(21)

The discretization for the advection equation (12) due to thisscheme (for 119906 gt 0) is given by

119902119899+1119894

= 119902119899

119894

minusΔ119905

Δ119909119906 [

5119902119894minus 119902119894minus1 + 2119902

119894+16

minus5119902119894minus1 minus 119902

119894minus2 + 2119902119894

6]

(22)

With algebraic manipulation it can be shown that thisscheme is equivalent to the scheme in (16) when the fluxlimiter function is chosen to be 120593 = 4(2 + 119903)3(3 + 119903) Hencewe define the darker shaded region in Figure 1 as the desired3rd-order TVD region for our scheme

To guarantee all the conditions we imposed are satisfiedwe derive a family of flux limiters lying in the 3rd-order TVDregion (darker shaded region) whenever possible Any fluxfunction in that region can be expressed as an arithmeticaverage of two limiter functions 120593 = 1 that corresponds tothe QUICK scheme and 120593 = 4(2 + 119903)3(3 + 119903) for the 3rd-order upwind scheme namely

120593lowast= 1+ 120575 (119903 minus 1)

3 (3 + 119903) 0 lt 120575 lt 1 (23)

Intersection points 119903lowast1and 119903lowast2are found to be 119903lowast

1= (9minus120575)(21minus

120575) and 119903lowast

2= (120575 + 15)(120575 + 3) (see Figure 2) We can see

that for any chosen value of 120575 the limiter function satisfiesour condition for smooth regions 120593lowast(1) = 1 The arithmeticaverage and the intersection points are shown in Figure 2

Based on the above we propose the following family offlux limiters characterized by the parameter 120575

120593 (119903) =

0 119903 lt 0

81199033 + 119903

0 lt 119903 lt9 minus 120575

21 minus 120575

1 + 120575 (119903 minus 1)3 (3 + 119903)

9 minus 120575

21 minus 120575lt 119903 lt

120575 + 15120575 + 3

83 + 119903

120575 + 15120575 + 3

lt 119903

(24)

120593 =4(2 + r)

3(3 + r)

2

18

16

14

12

1

08

06

04

02

00 15 21 25 3 35 4 5

Smoothness parameter (r)

Lim

iter f

unct

ion

120593 = 1

rlowast2

120593lowast = 1 +120575(r minus 1)

3(3 + r)

rlowast1

Figure 2 Arithmetically averaged limiter function

Equation (24) shows that the proposed limiter function iscontinuous and bounded for any value of the smoothnessparameter 119903 It also satisfies the condition for smooth regions120593(1) = 1 The flux limiter functions in (24) lie in the TVDregion for any choice of 120575 hence the scheme is TVD

In this paper the theta method (also called the gen-eralized Crank-Nicolson method) is used by consideringa weighted average of explicit and implicit terms in thenumerical scheme of (2) The scheme can be written as

119902119899+1119894

= 119902119899

119894minus120582 [120579 (F

119899+1119894+12 minusF

119899+1119894minus12)

+ (1minus 120579) (F119899119894+12 minusF

119899

119894minus12)] 120579 isin [0 1] (25)

Equation (25) represents a family of schemes characterizedby 120579 We obtain the explicit scheme for 120579 = 0 the implicitscheme for 120579 = 1 and the Crank-Nicolson scheme for 120579 =

12The thetamethod is unconditionally stable for any choiceof 120579 isin [12 1] [15 16] Spatially the accuracy of the schemeis determined by the way of calculating the numerical fluxesF119899119894plusmn12

4 Numerical Results and Discussion

In this section we present two sets of numerical resultsone for the case of smooth solutions and the other fordiscontinuous solutions Both sets of results are shown forthe case of pure advection with positive velocity Results fromthe proposed scheme are also compared to those from othernumerical schemes

41 Smooth Initial Conditions For this set of results weconsider (12) with a smooth initial condition given by

1199020 (119909) = 05+ 03 sin (2120587119909) (26)

Journal of Computational Engineering 5

0

05

1

0 05 1

0

05

1

005 1

0

05

1

0 05 1

0

05

1

0 05

1

1st-order upwind Lax-Wendroff

2nd-order upwind Minbee limiter

q q

q q

x

x

0

05

1

0 05 1

q

x

0

05

1

0 05

x

1

q

0

05

1

0 05

x

1

q

0

05

1

0 05 1

q

x

x

x

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 3 Numerical results for smooth solutions with 120579 = 1

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain Total timeof the simulation was set to 1 sec so that the solution at theend of the simulation is the same as the initial condition

Figure 3 shows a comparison of the new scheme withother classical schemes and existing high-resolution schemesfor the case of implicit time discretization (120579 = 1) Compari-son was made for a grid size of 005m and a Courant numberof 02

Excessive dissipation of the 1st-order upwind schemecan be observed and the solution is smeared out due tothe inherent numerical viscosity of the scheme For the twohigher order linear schemes (2nd-order upwind scheme and

Lax-Wendroff scheme) dissipative errors were minimizedbut the signal in the numerical solution was either leadingor lagging due to dispersion errors This dispersive nature ofthe error can be observed by studying the modified equationfor the corresponding schemes For example the modifiedequation corresponding to the 2nd-order upwind scheme is

120597119902

120597119905+ 119906

120597119902

120597119909= 119906

2Δ119905 (120579 minus

12)1205972119902

1205971199092

+(119906Δ119909

2

3minus1199063Δ119905

2

2(120579minus

13))

1205973119902

1205971199093

(27)

6 Journal of Computational Engineering

05

1

00

05 1

1st-order upwind

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

Lax-Wendroff

2nd-order upwind Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 4 Numerical results for smooth solutions with 120579 = 12

It can be seen from (27) that the 2nd-order upwind schemeposes two types of errors dissipation errors due to temporaldiscretization and dispersive errors due to both temporal andspatial discretization The latter type of error causes eitherleading or lagging of the numerical solution as shown inFigure 3

On the other hand more satisfactory results can beobserved when implementing high- resolution schemes withdifferent flux limitersThe new scheme with 120575 = 1 resulted ina better numerical solution as compared to other schemes

Figure 4 shows a comparison of the same set of schemesfor the case of Crank-Nicolson time discretization (120579 = 12)By examining (27) it is clear that the dissipative error termscease to exist for the case of 120579 = 12 This explains the

reduction in dissipation for the high-order linear schemes(2nd-order upwind and Lax-Wendroff) On the other handour new scheme continues to produce the best results ascompared to other schemes

The exact solution was used to determine the accuracy ofdifferent numerical schemes Accuracy was assessed on thebasis of the 119871

1norm defined as

1198711 =1119873

119873

sum

119894=1

1003816100381610038161003816119902119873 (119909119894) minus 119902119864 (119909119894)1003816100381610038161003816 (28)

where 119873 is the number of cells and 119902119873(119909119894) is the numerical

solution at the 119894th cell Convergence rate is represented bya log-log plot of the norm of the error versus mesh size

Journal of Computational Engineering 7

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(

erro

r)

1st-order upwindQUICK schemeMinbee limiter

Superbee limiterKumar limiter

log(dx)

Suggested limiter (delta = 1)

Figure 5 Spatial convergence rates for smooth solutions

Table 1 Spatial convergence rates of different schemes for smoothsolution

Numerical scheme Convergence rate1st-order upwind 086QUICK scheme 13Minbee limiter 11Superbee 099Kumar limiter 13New limiter with 120575 = 1 13

Figure 5 shows the results of the spatial convergence for thecase 120579 = 12 Convergence rates for several consideredschemes are listed in Table 1 The results show competitiverate of convergence for the new scheme as compared to otherschemes

42 Discontinuous Initial Conditions For this set of resultswe consider (12) with a discontinuous initial condition givenby

1199020 (119909) =

03 for 00 lt 119909 lt 04

08 for 04 lt 119909 lt 06

03 for 06 lt 119909 lt 10

(29)

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain and the timeof the simulation is 1 sec

Table 2 Spatial convergence rates of different schemes for discon-tinuous solution

Numerical scheme Convergence rate1st-order upwind 040Minbee limiter 056Superbee 052Kumar limiter 070Suggested limiter with 120575 = 1 082

Two sets of results are shown implicit time discretization(120579 = 1) is shown in Figure 6 and Crank-Nicolson timediscretization is shown in Figure 7 Comparison is made fora grid size of 00125m and Courant number of 02

Results show significantly more dissipation for the caseof implicit discretization For the 2nd-order upwind schemeand the QUICK scheme dispersion error appears in theformof overshoots and undershoots For theQUICK schemethese oscillations were damped for the case of 120579 = 1

due to dissipation High-resolution schemes including thenew scheme give better results capturing the discontinuityespecially for the case with 120579 = 12 where dissipation isreduced

Figure 8 shows results of the spatial convergence study fordifferent schemes Results of this study are listed in Table 2We can observe a very good rate of convergence for the newscheme compared to the other schemes

5 Concluding Remarks

A new high-resolution stable scheme is derived by imple-menting a hybridization of the monotone 1st-order upwindscheme and the quadratic upstream interpolation scheme(QUICK) For temporal discretization the generalizedCrank-Nicolson method characterized by the parameter 120579was used The combination of the 1st-order upwind schemeand QUICK scheme was done by means of a flux limitingfunction which depends on the smoothness of the solutionConstraints were imposed to make the resulting schemeTVD and high-order at smooth regions of the solution Forthe one-dimensional linear case results from the proposedscheme for different values of 120579 were compared to classicaland popular high-resolution schemes and the convergencerate for the scheme was investigated for smooth and dis-continuous solutions The proposed scheme was shown toexhibit very good results as compared to other schemes Itis shown that the high-resolution schemes based on 2ndorder Lax-Wendroff discretization exhibit a convergence rateof about 1 for smooth solutions and 05 for discontinuoussolutions The new scheme based on 3rd-order QUICKdiscretization exhibits a convergence rate of about 13 forsmooth solutions and 08 for discontinuous solutions In thecase of discontinuous solutions the new method derived inthis paper shows the best convergence rate of all schemesinvestigated here

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

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4 Journal of Computational Engineering

It is obvious that maximizing the flux limiter increasesthe antidiffusivity of the scheme This means that a fluxlimiter function corresponding to the upper boundary ofthe TVD region yields the least diffusive scheme possibleNevertheless choosing such a limiter does not guarantee thehighest possible order of accuracy To ensure high-order ofaccuracy we require one additional condition the limiterfunction should be chosen such that the scheme in (16) is 3rd-order accurate whenever possible To impose this constraintwe investigated another numerical scheme and the 3rd-orderupwind scheme For this scheme the numerical fluxes aredefined as

119902119880

119894+12 =

16(5119902119894minus 119902119894minus1 + 2119902

119894+1) 119906 gt 0

16(5119902119894+1 minus 119902

119894+2 + 2119902119894) 119906 lt 0

(21)

The discretization for the advection equation (12) due to thisscheme (for 119906 gt 0) is given by

119902119899+1119894

= 119902119899

119894

minusΔ119905

Δ119909119906 [

5119902119894minus 119902119894minus1 + 2119902

119894+16

minus5119902119894minus1 minus 119902

119894minus2 + 2119902119894

6]

(22)

With algebraic manipulation it can be shown that thisscheme is equivalent to the scheme in (16) when the fluxlimiter function is chosen to be 120593 = 4(2 + 119903)3(3 + 119903) Hencewe define the darker shaded region in Figure 1 as the desired3rd-order TVD region for our scheme

To guarantee all the conditions we imposed are satisfiedwe derive a family of flux limiters lying in the 3rd-order TVDregion (darker shaded region) whenever possible Any fluxfunction in that region can be expressed as an arithmeticaverage of two limiter functions 120593 = 1 that corresponds tothe QUICK scheme and 120593 = 4(2 + 119903)3(3 + 119903) for the 3rd-order upwind scheme namely

120593lowast= 1+ 120575 (119903 minus 1)

3 (3 + 119903) 0 lt 120575 lt 1 (23)

Intersection points 119903lowast1and 119903lowast2are found to be 119903lowast

1= (9minus120575)(21minus

120575) and 119903lowast

2= (120575 + 15)(120575 + 3) (see Figure 2) We can see

that for any chosen value of 120575 the limiter function satisfiesour condition for smooth regions 120593lowast(1) = 1 The arithmeticaverage and the intersection points are shown in Figure 2

Based on the above we propose the following family offlux limiters characterized by the parameter 120575

120593 (119903) =

0 119903 lt 0

81199033 + 119903

0 lt 119903 lt9 minus 120575

21 minus 120575

1 + 120575 (119903 minus 1)3 (3 + 119903)

9 minus 120575

21 minus 120575lt 119903 lt

120575 + 15120575 + 3

83 + 119903

120575 + 15120575 + 3

lt 119903

(24)

120593 =4(2 + r)

3(3 + r)

2

18

16

14

12

1

08

06

04

02

00 15 21 25 3 35 4 5

Smoothness parameter (r)

Lim

iter f

unct

ion

120593 = 1

rlowast2

120593lowast = 1 +120575(r minus 1)

3(3 + r)

rlowast1

Figure 2 Arithmetically averaged limiter function

Equation (24) shows that the proposed limiter function iscontinuous and bounded for any value of the smoothnessparameter 119903 It also satisfies the condition for smooth regions120593(1) = 1 The flux limiter functions in (24) lie in the TVDregion for any choice of 120575 hence the scheme is TVD

In this paper the theta method (also called the gen-eralized Crank-Nicolson method) is used by consideringa weighted average of explicit and implicit terms in thenumerical scheme of (2) The scheme can be written as

119902119899+1119894

= 119902119899

119894minus120582 [120579 (F

119899+1119894+12 minusF

119899+1119894minus12)

+ (1minus 120579) (F119899119894+12 minusF

119899

119894minus12)] 120579 isin [0 1] (25)

Equation (25) represents a family of schemes characterizedby 120579 We obtain the explicit scheme for 120579 = 0 the implicitscheme for 120579 = 1 and the Crank-Nicolson scheme for 120579 =

12The thetamethod is unconditionally stable for any choiceof 120579 isin [12 1] [15 16] Spatially the accuracy of the schemeis determined by the way of calculating the numerical fluxesF119899119894plusmn12

4 Numerical Results and Discussion

In this section we present two sets of numerical resultsone for the case of smooth solutions and the other fordiscontinuous solutions Both sets of results are shown forthe case of pure advection with positive velocity Results fromthe proposed scheme are also compared to those from othernumerical schemes

41 Smooth Initial Conditions For this set of results weconsider (12) with a smooth initial condition given by

1199020 (119909) = 05+ 03 sin (2120587119909) (26)

Journal of Computational Engineering 5

0

05

1

0 05 1

0

05

1

005 1

0

05

1

0 05 1

0

05

1

0 05

1

1st-order upwind Lax-Wendroff

2nd-order upwind Minbee limiter

q q

q q

x

x

0

05

1

0 05 1

q

x

0

05

1

0 05

x

1

q

0

05

1

0 05

x

1

q

0

05

1

0 05 1

q

x

x

x

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 3 Numerical results for smooth solutions with 120579 = 1

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain Total timeof the simulation was set to 1 sec so that the solution at theend of the simulation is the same as the initial condition

Figure 3 shows a comparison of the new scheme withother classical schemes and existing high-resolution schemesfor the case of implicit time discretization (120579 = 1) Compari-son was made for a grid size of 005m and a Courant numberof 02

Excessive dissipation of the 1st-order upwind schemecan be observed and the solution is smeared out due tothe inherent numerical viscosity of the scheme For the twohigher order linear schemes (2nd-order upwind scheme and

Lax-Wendroff scheme) dissipative errors were minimizedbut the signal in the numerical solution was either leadingor lagging due to dispersion errors This dispersive nature ofthe error can be observed by studying the modified equationfor the corresponding schemes For example the modifiedequation corresponding to the 2nd-order upwind scheme is

120597119902

120597119905+ 119906

120597119902

120597119909= 119906

2Δ119905 (120579 minus

12)1205972119902

1205971199092

+(119906Δ119909

2

3minus1199063Δ119905

2

2(120579minus

13))

1205973119902

1205971199093

(27)

6 Journal of Computational Engineering

05

1

00

05 1

1st-order upwind

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

Lax-Wendroff

2nd-order upwind Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 4 Numerical results for smooth solutions with 120579 = 12

It can be seen from (27) that the 2nd-order upwind schemeposes two types of errors dissipation errors due to temporaldiscretization and dispersive errors due to both temporal andspatial discretization The latter type of error causes eitherleading or lagging of the numerical solution as shown inFigure 3

On the other hand more satisfactory results can beobserved when implementing high- resolution schemes withdifferent flux limitersThe new scheme with 120575 = 1 resulted ina better numerical solution as compared to other schemes

Figure 4 shows a comparison of the same set of schemesfor the case of Crank-Nicolson time discretization (120579 = 12)By examining (27) it is clear that the dissipative error termscease to exist for the case of 120579 = 12 This explains the

reduction in dissipation for the high-order linear schemes(2nd-order upwind and Lax-Wendroff) On the other handour new scheme continues to produce the best results ascompared to other schemes

The exact solution was used to determine the accuracy ofdifferent numerical schemes Accuracy was assessed on thebasis of the 119871

1norm defined as

1198711 =1119873

119873

sum

119894=1

1003816100381610038161003816119902119873 (119909119894) minus 119902119864 (119909119894)1003816100381610038161003816 (28)

where 119873 is the number of cells and 119902119873(119909119894) is the numerical

solution at the 119894th cell Convergence rate is represented bya log-log plot of the norm of the error versus mesh size

Journal of Computational Engineering 7

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(

erro

r)

1st-order upwindQUICK schemeMinbee limiter

Superbee limiterKumar limiter

log(dx)

Suggested limiter (delta = 1)

Figure 5 Spatial convergence rates for smooth solutions

Table 1 Spatial convergence rates of different schemes for smoothsolution

Numerical scheme Convergence rate1st-order upwind 086QUICK scheme 13Minbee limiter 11Superbee 099Kumar limiter 13New limiter with 120575 = 1 13

Figure 5 shows the results of the spatial convergence for thecase 120579 = 12 Convergence rates for several consideredschemes are listed in Table 1 The results show competitiverate of convergence for the new scheme as compared to otherschemes

42 Discontinuous Initial Conditions For this set of resultswe consider (12) with a discontinuous initial condition givenby

1199020 (119909) =

03 for 00 lt 119909 lt 04

08 for 04 lt 119909 lt 06

03 for 06 lt 119909 lt 10

(29)

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain and the timeof the simulation is 1 sec

Table 2 Spatial convergence rates of different schemes for discon-tinuous solution

Numerical scheme Convergence rate1st-order upwind 040Minbee limiter 056Superbee 052Kumar limiter 070Suggested limiter with 120575 = 1 082

Two sets of results are shown implicit time discretization(120579 = 1) is shown in Figure 6 and Crank-Nicolson timediscretization is shown in Figure 7 Comparison is made fora grid size of 00125m and Courant number of 02

Results show significantly more dissipation for the caseof implicit discretization For the 2nd-order upwind schemeand the QUICK scheme dispersion error appears in theformof overshoots and undershoots For theQUICK schemethese oscillations were damped for the case of 120579 = 1

due to dissipation High-resolution schemes including thenew scheme give better results capturing the discontinuityespecially for the case with 120579 = 12 where dissipation isreduced

Figure 8 shows results of the spatial convergence study fordifferent schemes Results of this study are listed in Table 2We can observe a very good rate of convergence for the newscheme compared to the other schemes

5 Concluding Remarks

A new high-resolution stable scheme is derived by imple-menting a hybridization of the monotone 1st-order upwindscheme and the quadratic upstream interpolation scheme(QUICK) For temporal discretization the generalizedCrank-Nicolson method characterized by the parameter 120579was used The combination of the 1st-order upwind schemeand QUICK scheme was done by means of a flux limitingfunction which depends on the smoothness of the solutionConstraints were imposed to make the resulting schemeTVD and high-order at smooth regions of the solution Forthe one-dimensional linear case results from the proposedscheme for different values of 120579 were compared to classicaland popular high-resolution schemes and the convergencerate for the scheme was investigated for smooth and dis-continuous solutions The proposed scheme was shown toexhibit very good results as compared to other schemes Itis shown that the high-resolution schemes based on 2ndorder Lax-Wendroff discretization exhibit a convergence rateof about 1 for smooth solutions and 05 for discontinuoussolutions The new scheme based on 3rd-order QUICKdiscretization exhibits a convergence rate of about 13 forsmooth solutions and 08 for discontinuous solutions In thecase of discontinuous solutions the new method derived inthis paper shows the best convergence rate of all schemesinvestigated here

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Computational Engineering 5

0

05

1

0 05 1

0

05

1

005 1

0

05

1

0 05 1

0

05

1

0 05

1

1st-order upwind Lax-Wendroff

2nd-order upwind Minbee limiter

q q

q q

x

x

0

05

1

0 05 1

q

x

0

05

1

0 05

x

1

q

0

05

1

0 05

x

1

q

0

05

1

0 05 1

q

x

x

x

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 3 Numerical results for smooth solutions with 120579 = 1

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain Total timeof the simulation was set to 1 sec so that the solution at theend of the simulation is the same as the initial condition

Figure 3 shows a comparison of the new scheme withother classical schemes and existing high-resolution schemesfor the case of implicit time discretization (120579 = 1) Compari-son was made for a grid size of 005m and a Courant numberof 02

Excessive dissipation of the 1st-order upwind schemecan be observed and the solution is smeared out due tothe inherent numerical viscosity of the scheme For the twohigher order linear schemes (2nd-order upwind scheme and

Lax-Wendroff scheme) dissipative errors were minimizedbut the signal in the numerical solution was either leadingor lagging due to dispersion errors This dispersive nature ofthe error can be observed by studying the modified equationfor the corresponding schemes For example the modifiedequation corresponding to the 2nd-order upwind scheme is

120597119902

120597119905+ 119906

120597119902

120597119909= 119906

2Δ119905 (120579 minus

12)1205972119902

1205971199092

+(119906Δ119909

2

3minus1199063Δ119905

2

2(120579minus

13))

1205973119902

1205971199093

(27)

6 Journal of Computational Engineering

05

1

00

05 1

1st-order upwind

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

Lax-Wendroff

2nd-order upwind Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 4 Numerical results for smooth solutions with 120579 = 12

It can be seen from (27) that the 2nd-order upwind schemeposes two types of errors dissipation errors due to temporaldiscretization and dispersive errors due to both temporal andspatial discretization The latter type of error causes eitherleading or lagging of the numerical solution as shown inFigure 3

On the other hand more satisfactory results can beobserved when implementing high- resolution schemes withdifferent flux limitersThe new scheme with 120575 = 1 resulted ina better numerical solution as compared to other schemes

Figure 4 shows a comparison of the same set of schemesfor the case of Crank-Nicolson time discretization (120579 = 12)By examining (27) it is clear that the dissipative error termscease to exist for the case of 120579 = 12 This explains the

reduction in dissipation for the high-order linear schemes(2nd-order upwind and Lax-Wendroff) On the other handour new scheme continues to produce the best results ascompared to other schemes

The exact solution was used to determine the accuracy ofdifferent numerical schemes Accuracy was assessed on thebasis of the 119871

1norm defined as

1198711 =1119873

119873

sum

119894=1

1003816100381610038161003816119902119873 (119909119894) minus 119902119864 (119909119894)1003816100381610038161003816 (28)

where 119873 is the number of cells and 119902119873(119909119894) is the numerical

solution at the 119894th cell Convergence rate is represented bya log-log plot of the norm of the error versus mesh size

Journal of Computational Engineering 7

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(

erro

r)

1st-order upwindQUICK schemeMinbee limiter

Superbee limiterKumar limiter

log(dx)

Suggested limiter (delta = 1)

Figure 5 Spatial convergence rates for smooth solutions

Table 1 Spatial convergence rates of different schemes for smoothsolution

Numerical scheme Convergence rate1st-order upwind 086QUICK scheme 13Minbee limiter 11Superbee 099Kumar limiter 13New limiter with 120575 = 1 13

Figure 5 shows the results of the spatial convergence for thecase 120579 = 12 Convergence rates for several consideredschemes are listed in Table 1 The results show competitiverate of convergence for the new scheme as compared to otherschemes

42 Discontinuous Initial Conditions For this set of resultswe consider (12) with a discontinuous initial condition givenby

1199020 (119909) =

03 for 00 lt 119909 lt 04

08 for 04 lt 119909 lt 06

03 for 06 lt 119909 lt 10

(29)

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain and the timeof the simulation is 1 sec

Table 2 Spatial convergence rates of different schemes for discon-tinuous solution

Numerical scheme Convergence rate1st-order upwind 040Minbee limiter 056Superbee 052Kumar limiter 070Suggested limiter with 120575 = 1 082

Two sets of results are shown implicit time discretization(120579 = 1) is shown in Figure 6 and Crank-Nicolson timediscretization is shown in Figure 7 Comparison is made fora grid size of 00125m and Courant number of 02

Results show significantly more dissipation for the caseof implicit discretization For the 2nd-order upwind schemeand the QUICK scheme dispersion error appears in theformof overshoots and undershoots For theQUICK schemethese oscillations were damped for the case of 120579 = 1

due to dissipation High-resolution schemes including thenew scheme give better results capturing the discontinuityespecially for the case with 120579 = 12 where dissipation isreduced

Figure 8 shows results of the spatial convergence study fordifferent schemes Results of this study are listed in Table 2We can observe a very good rate of convergence for the newscheme compared to the other schemes

5 Concluding Remarks

A new high-resolution stable scheme is derived by imple-menting a hybridization of the monotone 1st-order upwindscheme and the quadratic upstream interpolation scheme(QUICK) For temporal discretization the generalizedCrank-Nicolson method characterized by the parameter 120579was used The combination of the 1st-order upwind schemeand QUICK scheme was done by means of a flux limitingfunction which depends on the smoothness of the solutionConstraints were imposed to make the resulting schemeTVD and high-order at smooth regions of the solution Forthe one-dimensional linear case results from the proposedscheme for different values of 120579 were compared to classicaland popular high-resolution schemes and the convergencerate for the scheme was investigated for smooth and dis-continuous solutions The proposed scheme was shown toexhibit very good results as compared to other schemes Itis shown that the high-resolution schemes based on 2ndorder Lax-Wendroff discretization exhibit a convergence rateof about 1 for smooth solutions and 05 for discontinuoussolutions The new scheme based on 3rd-order QUICKdiscretization exhibits a convergence rate of about 13 forsmooth solutions and 08 for discontinuous solutions In thecase of discontinuous solutions the new method derived inthis paper shows the best convergence rate of all schemesinvestigated here

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

6 Journal of Computational Engineering

05

1

00

05 1

1st-order upwind

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

Lax-Wendroff

2nd-order upwind Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 4 Numerical results for smooth solutions with 120579 = 12

It can be seen from (27) that the 2nd-order upwind schemeposes two types of errors dissipation errors due to temporaldiscretization and dispersive errors due to both temporal andspatial discretization The latter type of error causes eitherleading or lagging of the numerical solution as shown inFigure 3

On the other hand more satisfactory results can beobserved when implementing high- resolution schemes withdifferent flux limitersThe new scheme with 120575 = 1 resulted ina better numerical solution as compared to other schemes

Figure 4 shows a comparison of the same set of schemesfor the case of Crank-Nicolson time discretization (120579 = 12)By examining (27) it is clear that the dissipative error termscease to exist for the case of 120579 = 12 This explains the

reduction in dissipation for the high-order linear schemes(2nd-order upwind and Lax-Wendroff) On the other handour new scheme continues to produce the best results ascompared to other schemes

The exact solution was used to determine the accuracy ofdifferent numerical schemes Accuracy was assessed on thebasis of the 119871

1norm defined as

1198711 =1119873

119873

sum

119894=1

1003816100381610038161003816119902119873 (119909119894) minus 119902119864 (119909119894)1003816100381610038161003816 (28)

where 119873 is the number of cells and 119902119873(119909119894) is the numerical

solution at the 119894th cell Convergence rate is represented bya log-log plot of the norm of the error versus mesh size

Journal of Computational Engineering 7

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(

erro

r)

1st-order upwindQUICK schemeMinbee limiter

Superbee limiterKumar limiter

log(dx)

Suggested limiter (delta = 1)

Figure 5 Spatial convergence rates for smooth solutions

Table 1 Spatial convergence rates of different schemes for smoothsolution

Numerical scheme Convergence rate1st-order upwind 086QUICK scheme 13Minbee limiter 11Superbee 099Kumar limiter 13New limiter with 120575 = 1 13

Figure 5 shows the results of the spatial convergence for thecase 120579 = 12 Convergence rates for several consideredschemes are listed in Table 1 The results show competitiverate of convergence for the new scheme as compared to otherschemes

42 Discontinuous Initial Conditions For this set of resultswe consider (12) with a discontinuous initial condition givenby

1199020 (119909) =

03 for 00 lt 119909 lt 04

08 for 04 lt 119909 lt 06

03 for 06 lt 119909 lt 10

(29)

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain and the timeof the simulation is 1 sec

Table 2 Spatial convergence rates of different schemes for discon-tinuous solution

Numerical scheme Convergence rate1st-order upwind 040Minbee limiter 056Superbee 052Kumar limiter 070Suggested limiter with 120575 = 1 082

Two sets of results are shown implicit time discretization(120579 = 1) is shown in Figure 6 and Crank-Nicolson timediscretization is shown in Figure 7 Comparison is made fora grid size of 00125m and Courant number of 02

Results show significantly more dissipation for the caseof implicit discretization For the 2nd-order upwind schemeand the QUICK scheme dispersion error appears in theformof overshoots and undershoots For theQUICK schemethese oscillations were damped for the case of 120579 = 1

due to dissipation High-resolution schemes including thenew scheme give better results capturing the discontinuityespecially for the case with 120579 = 12 where dissipation isreduced

Figure 8 shows results of the spatial convergence study fordifferent schemes Results of this study are listed in Table 2We can observe a very good rate of convergence for the newscheme compared to the other schemes

5 Concluding Remarks

A new high-resolution stable scheme is derived by imple-menting a hybridization of the monotone 1st-order upwindscheme and the quadratic upstream interpolation scheme(QUICK) For temporal discretization the generalizedCrank-Nicolson method characterized by the parameter 120579was used The combination of the 1st-order upwind schemeand QUICK scheme was done by means of a flux limitingfunction which depends on the smoothness of the solutionConstraints were imposed to make the resulting schemeTVD and high-order at smooth regions of the solution Forthe one-dimensional linear case results from the proposedscheme for different values of 120579 were compared to classicaland popular high-resolution schemes and the convergencerate for the scheme was investigated for smooth and dis-continuous solutions The proposed scheme was shown toexhibit very good results as compared to other schemes Itis shown that the high-resolution schemes based on 2ndorder Lax-Wendroff discretization exhibit a convergence rateof about 1 for smooth solutions and 05 for discontinuoussolutions The new scheme based on 3rd-order QUICKdiscretization exhibits a convergence rate of about 13 forsmooth solutions and 08 for discontinuous solutions In thecase of discontinuous solutions the new method derived inthis paper shows the best convergence rate of all schemesinvestigated here

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Computational Engineering 7

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus7

minus65

minus6

minus55

minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(

erro

r)

1st-order upwindQUICK schemeMinbee limiter

Superbee limiterKumar limiter

log(dx)

Suggested limiter (delta = 1)

Figure 5 Spatial convergence rates for smooth solutions

Table 1 Spatial convergence rates of different schemes for smoothsolution

Numerical scheme Convergence rate1st-order upwind 086QUICK scheme 13Minbee limiter 11Superbee 099Kumar limiter 13New limiter with 120575 = 1 13

Figure 5 shows the results of the spatial convergence for thecase 120579 = 12 Convergence rates for several consideredschemes are listed in Table 1 The results show competitiverate of convergence for the new scheme as compared to otherschemes

42 Discontinuous Initial Conditions For this set of resultswe consider (12) with a discontinuous initial condition givenby

1199020 (119909) =

03 for 00 lt 119909 lt 04

08 for 04 lt 119909 lt 06

03 for 06 lt 119909 lt 10

(29)

Boundary conditions for this problem are periodic with aconvection velocity 119906 = 1ms on a 1m domain and the timeof the simulation is 1 sec

Table 2 Spatial convergence rates of different schemes for discon-tinuous solution

Numerical scheme Convergence rate1st-order upwind 040Minbee limiter 056Superbee 052Kumar limiter 070Suggested limiter with 120575 = 1 082

Two sets of results are shown implicit time discretization(120579 = 1) is shown in Figure 6 and Crank-Nicolson timediscretization is shown in Figure 7 Comparison is made fora grid size of 00125m and Courant number of 02

Results show significantly more dissipation for the caseof implicit discretization For the 2nd-order upwind schemeand the QUICK scheme dispersion error appears in theformof overshoots and undershoots For theQUICK schemethese oscillations were damped for the case of 120579 = 1

due to dissipation High-resolution schemes including thenew scheme give better results capturing the discontinuityespecially for the case with 120579 = 12 where dissipation isreduced

Figure 8 shows results of the spatial convergence study fordifferent schemes Results of this study are listed in Table 2We can observe a very good rate of convergence for the newscheme compared to the other schemes

5 Concluding Remarks

A new high-resolution stable scheme is derived by imple-menting a hybridization of the monotone 1st-order upwindscheme and the quadratic upstream interpolation scheme(QUICK) For temporal discretization the generalizedCrank-Nicolson method characterized by the parameter 120579was used The combination of the 1st-order upwind schemeand QUICK scheme was done by means of a flux limitingfunction which depends on the smoothness of the solutionConstraints were imposed to make the resulting schemeTVD and high-order at smooth regions of the solution Forthe one-dimensional linear case results from the proposedscheme for different values of 120579 were compared to classicaland popular high-resolution schemes and the convergencerate for the scheme was investigated for smooth and dis-continuous solutions The proposed scheme was shown toexhibit very good results as compared to other schemes Itis shown that the high-resolution schemes based on 2ndorder Lax-Wendroff discretization exhibit a convergence rateof about 1 for smooth solutions and 05 for discontinuoussolutions The new scheme based on 3rd-order QUICKdiscretization exhibits a convergence rate of about 13 forsmooth solutions and 08 for discontinuous solutions In thecase of discontinuous solutions the new method derived inthis paper shows the best convergence rate of all schemesinvestigated here

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

8 Journal of Computational Engineering

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 6 Numerical results for discontinuous solutions with 120579 = 1

Appendix

Finding the TVD Region for a NegativeVelocity (119906 lt 0)

The numerical scheme is given by

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119891

119894+12minus 119902119891

119894minus12

+120593119894+12 [119902

119904

119894+12 minus 119902119891

119894+12] minus 120593119894minus12 [119902119904

119894minus12 minus 119902119891

119894minus12]]

(A1)

From (13) and (14) we substitute the numerical fluxes to get

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909119906 [119902119894+1 minus 119902119894

+18120593119894+12 [3119902119894 minus 3119902119894+1 + 119902119894+1 minus 119902119894+2]

minus18120593119894minus12 [minus3119902119894 + 3119902119894minus1 + 119902119894 minus 119902119894+1]] = 119902

119899

119894minusΔ119905

Δ119909

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Journal of Computational Engineering 9

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00 05 1

q

x

05

1

00 05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

05

1

00

05 1

q

x

1st-order upwind 2nd-order upwind

QUICK scheme Minbee limiter

Vanleer limiter Superbee limiter

Kumar limiter

NumericalExact

NumericalExact

Suggested scheme with delta = 1

Figure 7 Numerical results for discontinuous solutions with 120579 = 12

sdot 119906 [1

minus120593119894+12

8[3+

(119902119894+2 minus 119902

119894+1)

(119902119894+1 minus 119902

119894)]

+120593119894minus12

8[3

(119902119894minus 119902119894minus1)

(119902119894+1 minus 119902

119894)+ 1]] (119902

119894+1 minus 119902119894)

(A2)

Defining the smoothness parameter 119903119894= (119902119894+2

minus 119902119894+1)(119902119894+1

minus

119902119894) the above equation can be written as

119902119899+1119894

= 119902119899

119894minusΔ119905

Δ119909

sdot 119906 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

sdot (119902119894+1 minus 119902119894)

(A3)

This is an incremental form of the numerical scheme with

119862119894minus12 = 0

119863119894+12

= minusΔ119905

Δ119909119906 [1minus

120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]]

(A4)

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

10 Journal of Computational Engineering

minus6 minus55 minus5 minus45 minus4 minus35 minus3 minus25minus5

minus45

minus4

minus35

minus3

minus25

minus2

log(dx)

1st-order upwindMinbee limiterSuperbee limiter

Kumar limiter

log(L1)

Suggested limiter (delta = 0)

Figure 8 Spatial convergence rates for discontinuous solutions

Let 119888 = minus119906Δ119905Δ119909 gt 0 we implement conditions in (10) to get

0 le 119888 [1minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1]] le 1

minus 1 le minus120593119894+12

8[3+ 119903119894] +

120593119894minus12

8[

3119903119894minus1

+ 1] le1 minus 119888

119888

minus 8 le 120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

le 8(1 minus 119888

119888)

(A5)

For 119888 le 1210038161003816100381610038161003816100381610038161003816120593119894minus12 [

3119903119894minus1

+ 1]minus120593119894+12 [3+ 119903119894]

10038161003816100381610038161003816100381610038161003816le 8 (A6)

One condition we require for the limiter function is

120593 (119903) = 0 for 119903 lt 0 (A7)

This condition yields the following

0 le (120593119894minus12 [

3119903119894minus1

+ 1] 120593119894+12 [3+ 119903119894]) le 8 (A8)

From which we obtain the two inequalities

0 le 120593119894minus12 le [

8119903119894minus1

3 + 119903119894minus1

] 997888rarr 0 le 120593 le8119903

3 + 119903

0 le 120593119894+12 le [

83 + 119903119894

] 997888rarr 0 le 120593 le [8

3 + 119903]

(A9)

This is the same result as found for the case of a positivevelocity

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Harten ldquoHigh resolution schemes for hyperbolic conserva-tion lawsrdquo Journal of Computational Physics vol 135 pp 260ndash278 1997

[2] P K Sweby ldquoHigh resolution schemes using flux limiters forhyperbolic conservation lawsrdquo SIAM Journal on NumericalAnalysis vol 21 no 5 pp 995ndash1011 1984

[3] M K Kadalbajoo and R Kumar ldquoA high resolution totalvariation diminishing scheme for hyperbolic conservation lawand related problemsrdquo Applied Mathematics and Computationvol 175 no 2 pp 1556ndash1573 2006

[4] A Harten and S Osher ldquoUniformly high-order accuratenonoscillatory schemesrdquo SIAM Journal on Numerical Analysisvol 24 no 2 pp 279ndash309 1987

[5] A Harten B Engquist S Osher and S R Chakravarthy ldquoUni-formly high order accurate essentially non-oscillatory schemesIIIrdquo Journal of Computational Physics vol 71 no 2 pp 231ndash3031987

[6] X-D Liu S Osher and T Chan ldquoWeighted essentially non-oscillatory schemesrdquo Journal of Computational Physics vol 115no 1 pp 200ndash212 1994

[7] R J Leveque Finite Volume Methods for Hyperbolic ProblemsCambridgeUniversity Press Cambridge UK 10th edition 2011

[8] E F Toro Riemann Solvers and Numerical Methods for FluidDynamics A Practical Introduction Springer Dordrecht TheNetherlands 3rd edition 2009

[9] B van Leer ldquoTowards the ultimate conservative differencescheme V A second-order sequel to Godunovrsquos methodrdquoJournal of Computational Physics vol 32 no 1 pp 101ndash136 1979

[10] P Colella and P R Woodward ldquoThe Piecewise ParabolicMethod (PPM) for gas-dynamical simulationsrdquo Journal ofComputational Physics vol 54 no 1 pp 174ndash201 1984

[11] P K Sweby and M J Baines ldquoOn convergence of Roersquos schemefor the general nonlinear scalar wave equationrdquo Journal ofComputational Physics vol 56 no 1 pp 135ndash148 1984

[12] S K Godunov ldquoA difference method for the numerical calcu-lation of discontinuous solution of hydrodynamic equationsrdquoMatematicheskii Sbornik vol 47 pp 271ndash306 1957 Translatedas JPRS 7225 by US Department of Commerce pages 36 41 and106 1960

[13] P Lax and B Wendroff ldquoSystems of conservation lawsrdquo Com-munications on Pure and Applied Mathematics vol 13 no 2 pp217ndash237 1960

[14] B P Leonard ldquoA stable and accurate convective modeling pro-cedure based on quadratic upstream interpolationrdquo ComputerMethods in AppliedMechanics and Engineering vol 19 no 1 pp59ndash98 1979

[15] D J Higham ldquoMean-square and asymptotic stability of thestochastic theta methodrdquo SIAM Journal on Numerical Analysisvol 38 no 3 pp 753ndash769 2000

[16] I Farago ldquoConvergence and stability constant of the theta-methodrdquo in Proceedings of the Conference of Applications ofMathematics Prague Czech Republic 2013

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

RotatingMachinery

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Shock and Vibration

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Civil EngineeringAdvances in

Acoustics and VibrationAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawi Publishing Corporation httpwwwhindawicom

Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of