high resolution wavelet based central schemes for modeling nonlinear propagating fronts ·...

HIGH RESOLUTION WAVELET BASED CENTRAL SCHEMES

FOR MODELING NONLINEAR PROPAGATING FRONTS

HASSAN YOUSEFI, EITAN TADMOR, AND TIMON RABCZUK

Abstract. A class of wavelet-based methods, which combine adaptive grids together with second-order central and central-upwind high-resolution schemes, is introduced for accurate solution of firstorder hyperbolic systems of conservation laws and related equations. It is shown that the proposed

class recovers stability which otherwise may be lost when the underlying central schemes are utilizedover non-uniform grids. For simulations on irregular grids, both full-discretized and semi-discretizedforms are derived; in particular, the effect of using certain shifts of the center of the computational

allow to use standard slope limiters on non-uniform cells. Thereafter, the questions of numericalentropy production, local truncation errors, and the Total Variation Diminishing (TVD) criterion forscalar equations, are investigated.

Central schemes are sensitives for irregular grids. To cure this drawback in the present context

of multiresolution, the adapted grid is locally modified around high-gradient zones by adding newneighboring points. For an adapted point of resolution scale j, the new points are locally added inboth resolution j and all successive coarser resolutions. It is shown that this simple grid modificationimproves stability of numerical solutions; this, however, comes at the expense that cell-centered central

and central-upwind high resolution schemes, may not satisfy the TVD property. Finally, we presentnumerical simulations for both scalar and systems of non-linear conservation laws which confirm thesimplicity and efficiency of the proposed method. These simulations demonstrate the high accuracy,the entropy production of wavelet-based algorithms which can effectively detect high gradient zones

of shock waves, rarefaction regions, and contact discontinuities.

1. Introduction

Multiresolution-based studying has rapidly been developed in many branches of science; one of suchpowerful schemes is the wavelet theory. Wavelets act as a mathematical microscope and can detectabrupt variations. Development of this theory is simultaneously done by scientists, mathematiciansand engineers [50]. Wavelets can detect different local features of data separated locally in differentresolutions. Wavelets can efficiently distinguish overall smooth variation of a solution from locallyhigh transient ones. This multiresolution feature has interested many researchers in the field of partialdifferential equations (PDEs) [50]. One approach is to study problems in accordance with their solutionvariations; i.e., using different accuracies in different computational domains. In this approach, moregrid points are concentrated around high-gradient zones to detect high variations. Such simulationslead to adaptive solvers. In this case, only the important physics of problems are precisely studied fora cost-effective computations.

In general, wavelet-based adaptive methods have successfully been implemented for PDE solutionscontaining steep moving fronts or sharp transitions in small zones [11, 12, 1, 20, 47, 2, 55, 5, 37, 56, 58,57, 6, 2, 10, 25, 26, 9, 38, 39, 18, 13, 23, 15]. In most of these approaches, wavelets are used as a tool todetect localized spatial behaviors and corresponding zones. Wavelet coefficients of considerable valuesconcentrate automatically in the vicinity of high-gradient regions. The wavelet coefficients have a oneto one correspondence with corresponding spatial grid points. Hence, by ignoring coefficients of smallvalues and coresponding grid points, a considered grid can be adapted. In these grid-based adaptiveschemes, the degrees of freedom are considered as point values in the physical space [11, 12, 26].

In this work, propagation of nonlinear fronts are simulated by central high resolution schemes withproper integration of them with wavelet-based adaptation procedures. Accuracy and effectiveness of

Date: May 6, 2015.1991 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.Key words and phrases. Interpolating wavelet, adapted grid, central high-resolution schemes, the KT method, nonlin-

ear waves, discontinuous fronts.

1

2 HASSAN YOUSEFI, EITAN TADMOR, AND TIMON RABCZUK

numerical solutions can be obtained by such incorporation. In numerical simulation of first orderhyperbolic systems, several important and challenging features exist, as: lack of inherent (natural)dissipation [7], forming of artificial (numerical) dissipation and dispersion [3, 8]. Inherent dissipation ina system improves both adaptation procedure and numerical stability. The numerical dispersion leadsto developing wiggles in front and behind of propagating waves, due to distortion of different phases ofpropagating waves (fronts). The numerical dissipation has a tendency to flat discontinuities in numericalsolutions [3, 8]. Discontinuous solutions are commonly formed in non-linear first-order hyperbolicsystems, where controlling both the numerical dissipation and dispersion are challenging. Despite ofhyperbolic systems, the inherent dissipation exists in elliptic and parabolic PDEs [19]. These systemsare not so sensitive to small perturbations in their numerical solutions. The perturbations or errorsdissipate during numerical simulations. Based on this feature, wavelet-based adaptive schemes havesuccessfully been employed in modeling of the elliptic [5, 55] and parabolic problems [11, 12, 1, 37, 26].

Considering hyperbolic problems, however, handling the artificial effects -dissipation and dispersion,needs special treatment. Significant non-physical oscillations would strongly form around the (nearly)discontinuous solutions: the Gibb’s phenomenon. Due to such oscillations, numerical instability couldoccur. These spurious oscillations could propagate throughout the computational domain. Therefore,both the proper adaptation procedure and stability of numerical solutions can fail.

To guarantee solution stability of high-resolution schemes on non-uniform grids, some conditions mustbe satisfied. Such assurance can be obtained by using the normalized variable and space formulation(NVSF) criterion [14]. This condition is successfully incorporated with wavelet-based adaptive methodsfor simulation of hyperbolic systems [47, 2]. In the NVSF criterion, identifying of propagating directionsare necessary. This identification needs itself complex procedures, especially for 2-D and 3-D problems[47]. To remedy this disadvantage, in this work, central and central-upwind high resolution schemes areconsidered (like the Kurganov and Tadmor method) [32, 30, 29]. Central/central-upwind high resolu-tion schemes on non-staggered grids offer the following benefits: having a simple and straightforwardconcept; being easy to implement; having less numerical dissipation than ones on the staggered cells, likethe Nessyahu and Tadmor (NT) method; offering both semi-discrete and fully-discrete forms; being aRiemann-free solver; no requiring to staggered grid points as needed in the NT scheme [40]; having com-parable second and higher order accuracies with other expensive techniques. Central/central-upwindmethods, however, are sensitive for cell irregularity; our investigations show that numerical instabili-ties appear rapidly in adaptive solutions. This is because, these methods do not originally satisfy theNVSF condition. To guarantee the numerical stability, two features should be studied: 1) performanceof slope/flux limiters on non-uniform grids; 2) effects of grid density variation on numerical solutionsand adaptation procedures. Most of slope limiters have been developed for working properly on uni-form grids. Using of such limiters on irregular grids leads to unstable solutions. Abrupt changing ofgrid densities can also lead to instability, due to ill-posedness feature of irregular sampled data. Toprevent this kind of instability, density variation of adapted grids should be checked. For achievingthis purpose, adapted grids are locally modified in the vicinity of high-gradient zones to achieve localsemi-uniform grid points (then an ill-posed problem is nearly replaced with a well-posed one). Thismodification is done in context of the multiresolution analysis by a post-processing stage. Consideringan adapted grid, for each point of level resolution j0, some extra new neighbor points are locally added.The new points are inserted in both the same resolution level ( j0) and all successive coarser resolutionlevels, i.e.: j ∈ j0 − 1, j0 − 2, . . . , Jmin. It will be shown that even-though grid modification improvesnumerical stability, there is no guarantee to have long-term stability. This is due to performance ofslope/flux limiters on non-uniform cells. For non-uniform cells, one approach is to re-design limiters[4, 59]. In this study, it will be shown how to use limiters without their definition modification. Thiswill be done by re-locating cell centers only in some cells acting as transmitting cells. Investigationsshow that the proposed method leads to stable results comparable with those of NVSF-based schemes.

In nonlinear conservation laws, discontinuous solutions develop typically. Thereby, common errorestimation concepts, mainly based on the Taylor expansion, can not be used. For this reason, conceptof the local truncation error is used to assess convergence of solutions to weak ones as a practicalapproach [28, 27]. Related formulations for such error estimation would be provided for 1-D and 2-D

non-uniform cells. This kind of error has direct relationship with the Lip′-norm theory introduced by

Tadmor [52, 53] for nonlinear 1-D scalar conservation laws with convex flux. The numerical results

HIGH RESOLUTION WAVELET BASED CENTRAL SCHEMES 3

confirm that the local truncation error can also be used for convergence study of 1-D systems or 2-Dconservation laws, even with non-convex fluxes.

Uniqueness of numerical solutions is checked by the concept of numerical entropy production. Theo-retically, the numerical entropy production is zero in smooth regions while less than zero around shocksand discontinuities [43, 44]. This helps to study quality of numerical results especially for ones withoutexact solutions. All calculations will directly be done on non-uniform grids.

Concepts of the local truncation error and numerical entropy production have been used for bothgrid and method adaptation [31, 45, 17]. Different concepts could lead to different adapted results,especially, some concepts may not capture some phenomena. Hence, proper choosing of a adaptationapproach would be crucial. In this study, wavelets are used for grid/method adaptation. Adaptationperformance of this theory would be compared with above-mentioned two concepts.

Hyperbolic systems with non-convex fluxes would also be studied. These systems can explain im-portant phenomena, such as: Euler equations of gas dynamic with a non-convex flux, polymer systemused for simulation of polymer flooding processes in enhanced oil recovery and mechanical wave equa-tions with non-convex fluxes. It will be shown that even though numerical solutions converge to weakform solutions (controlled by the local truncation error), they may not be physical (real) ones due toexistence of complex waves in these problems [31]. In this work, wavelets are used to both grid andmethod adaptations for capturing properly physical solutions.

The Dubuc-Deslauriers (D-D) interpolating wavelets [11, 12, 36] are used for grid adaptation. Thisfamily has simple and straightforward algorithms with physical meaning. All calculations can then bedone in the physical domain. The D-D wavelets use minimal spatial support for data reconstruction(approximation), and this is important, since: larger inter-distance in two sampled data is, smallercorrelation between them exists [11, 12, 26].

Semi-discretized PDEs in spatial domain are solved in time by an explicit TVD integration method,such as the second order TVD Runge-Kutta scheme. As all spatio-temporal calculations are done inthe physical domain, the method is simple and conceptually straightforward [26].

This paper is composed of ten parts. Section 2: explaining the main concept of multiresolution-based grid adaptation by interpolating wavelets; Section 3: presenting post-processing of adapted girds;Section 4: explaining the main concept of central or central-upwind scheme with cell-centered cell points;Section 5: evaluating of full-discrete and semi-discrete forms of conservation laws on non-uniform cellswith shifted cell-centered points; section 6: estimation of the numerical entropy production; section 7:evaluating of local truncation errors on 1-D and 2-D non-uniform grids; section 8: explanation of theTVD conditions on non-uniform grids; section 9: representing some 1-D and 2-D numerical examples.The conclusion is presented at the end of the paper.

2. Multiresolution analysis and adaptation of 1-D grids

2.1. Multiresolution representation of 1-D grids. A dyadic grid on spatial interval [0, 1] is assumedas follows [12]:

Vj =

xj ,k ∈ [0, 1] : xj,k =

k

2jfor all j ∈ Z, and k ∈ 0, 1, ..., 2j

,(2.1)

where j and k are the resolution level (corresponding to spatial scale 1/2j) and spatial position (k/2j),respectively. This definition of dyadic grid points Vj in Eq. (2.1) leads to the condition xj−1,k = xj,2kand the multiresolution representation core: i.e., Vj ⊂ Vj+1. The points belongs to Vj+1 \Vj is denotedby Wj , and it can be expressed as:

Wj =

xj+1,2k+1 ∈ (0, 1) : xj+1,2k+1 =

2k + 1

2j+1for all j ∈ Z, and k ∈ 0, 1, ..., 2j − 1

.

So it can intuitively be concluded that: Vj ⊕Wj = Vj+1. This means the detail subspace Wj with theapproximation subspace Vj can create (span) the next finer approximation subspace Vj+1 with moredetails. By repeating this decomposition procedure on VJmax , it is obvious that:

VJmax = VJmin ⊕Nd∑i=0

WJmin+i, Nd = Jmax − Jmin − 1,


where, Jmax, Jmin and Nd denote the finest resolution, the coarsest resolution and number of decom-posing levels, respectively [36].

A continuous function f(x), defined on VJmax , is assumed (i.e., x ∈ VJmax). Regarding the multires-olution representation, the function can be decomposed as [12, 36]:

f(x) =2Jmin∑l=0

cJmin,lϕJmin,l(x) +

Jmax−1∑j=Jmin

2j−1∑n=0

dj,nψj,n(x)

= Pf Jmin+

Jmax−1∑j=Jmin

Qf j ,

where ϕ(x) and ψ(x) are scaling and wavelet functions, respectively; sets ϕj,k and ψj,l denote dilatedand shifted versions of ϕ(x) and ψ(x), respectively. Coefficients cj,l and dj,k are respectively approxi-mation and detail coefficients with resolution j. The operators Pf j and Qf j show the approximationand detail information of f(x), defined on grid points Vj , and Wj , respectively. The approximation onsuccessive finer resolution j + 1, can then be obtained as: Pf j+1 = Pf j +Qf j [36].

In this study, the interpolating D-D wavelet of order 2M−1 with support Supp(ϕ) = [−2M+1, 2M−1]is used. It can be obtained by auto-correlation of the Daubechies scaling function of orderM (havingMvanishing moments) [36]. By using this family, the transform coefficients in Eq. (2.2) can totally beevaluated in the physical space with physical meanings. The approximation coefficients (cJmin,l) areequal to sampled values of f(x) at points xJmin,l ∈ VJmin , and the detail coefficients (dj,n) are differencesbetween f(xj+1,2n+1) and Pf j+1(xj+1,2n+1), or dj,n = f(xj+1,2n+1) − Pf j+1(xj+1,2n+1) (for the D-Dwavelets, Pf j+1(xj+1,2n+1) can be obtained by the local Lagrange interpolation on points Vj); for detailssee [11, 12, 36].

In the interpolating wavelet theory, transform coefficients and grid points have one-to-one corre-spondence. This feature leads to a simple 1-D grid adaptation algorithm. For f(x) ∈ VJmax , a pre-defined threshold ϵ is assumed, then, in each level of resolution j ∈ Jmin, Jmin + 1, · · · , Jmax − 1 ,points xj+1,2n+1 ∈ Wj are omitted from original calculating grid points, if corresponding detail coef-ficients, dj,n satisfy the condition dj,n < ϵ. This means that the function at the point xj+1,2n+1 issmooth enough, so that its contribution in the approximation, dj,nψj,n(x), can be neglected (see Eq.(2.2)). Donoho [16] showed that such truncation error is in accordance with threshold values. Finally itshould be mentioned that the predefined threshold can be either level-dependent or not. One popularlevel-dependent threshold is ϵj = ϵ0/2

Jmax−j−1 for j ∈ Jmin, Jmin + 1, · · · Jmax − 1. This means asmaller threshold is assumed for finer resolution. To use a constant threshold value for all resolutionlevels, it is better to normalize the detail coefficient of resolution j as: dj,n = dj,n/f

refj . In this study

constant pre-defined threshold with normalized factor frefj = maxf(xj) is used.The presented adaptation procedure was shown to be efficient in the resolution of scalar functions.

For resolution of functions in vector system, the previous procedure is modified to reflect the solutions’behaviors of all equations. Namely, the resultant adapted grid is simply superposition of all adaptedgrids; which each adapted grid corresponds to a functions of the vector system.

2.2. Multiresolution representation of 2-D grids. Consider a uniform grid of spatial locations (x, y) ∈[0, 1]× [0, 1]; a set of points belonging to subspace Vj can be defined as [47]:

Vj =(xj,k, yj,l) : xj,k = k2−j , yj,l = l2−j

for all j, k, l ∈ Z,(2.2)

where j denotes resolution level and Jmin ≤ j ≤ Jmax; coefficients k & l measure spatial locations.Figure 1 illustrates a schematic representation of points belonging to Vj (• ∈ Vj) and Vj+1 (•∪ ∈ Vj+1)for case j = 2.

Same as the 1-D case, for 2-D information a detail subspace Wj belonging to subspace Vj+1 \ Vj canbe defined. These points are shown in Figure 1 with hollow circles; i.e.: ∈Wj . In this detail sub-space,three point types s1, s2 and s3 can be distinguished, as:

(1) points in even-numbered points in the x direction and odd-numbered in the y direction; i.e.,set s1 = (xj+1,2k, yj+1,2l+1),


s2

s1s3

Figure 1. Spatial locations of points belonging to subspaces V2 and V3, where • ∈V2 and • ∪ ∈ V3, for (xi, yj) ∈ [0, 1]× [0, 1].

(2) points in odd-numbered points in the x direction and even-numbered in the y direction; i.e.,set s2 = (xj+1,2k+1, yj+1,2l),

(3) odd-numbered grid points in both directions: s3 = (xj+1,2k+1, yj+1,2l+1).Based on subspaces Vj and Wj , a 2-D multiresolution analysis can be obtained as the 1-D case.

For point sets (xj,k, yj,l) ∈ si : i ∈ 1, 2, 3, four wavelet coefficientsdij,k,l : i ∈ 1, 2, 3, 4

can be

defined as:

(1) For points s1, wavelet coefficients are d1j,k,l; they measure local variations in vertical (y)

direction,(2) For points s2, wavelet coefficients are d2j,k,l; they measure local variations in horizontal (x)

direction,(3) For points s3, two sets of wavelet coefficients can be defined as d3j,k,l and d4j,k,l; they measure

respectively local variations in x and y directions.

These coefficients can be evaluated by the 1-D wavelet transform algorithm [47]. For 2-D grid adapta-tion, by considering a threshold value, grid points can then be adapted as the 1−D case [47].

3. Post processing the adapted grid points

Once adapted grids are obtained via the aforementioned wavelet-based procedures, grids are modifiedconsidering resolution level of each point; this is done to guarantee gradual density variation of gridpoints.

3.1. 1-D grid modification. The procedure for modification of 1-D adapted grids can be summarizedas:

(1) Setting the level resolution (j) equal to the finest resolution, i.e.: j = Jmax − 1,(2) Considering points belonging to the detail space of resolution j; i.e., points:

xj+1,2k+1 = (2k + 1)/2j+1 ∈Wj

,

(3) Existence controlling of Ns neighbor-points for each side of the point xj,k ∈ Wj at the samelevel of resolution, i.e., points:xj+1,2(k+i)+1 : i ∈ −Ns,−Ns + 1, · · · , Ns , i = 0

,

(4) Existence controlling of Nc neighbor-points for each side of the point xj,k ∈Wj at the successivecoarser resolution (subspace Wj−1). This step is only done for levels j > Jmin,

(5) Adding the extra points from steps 3 and 4 to the adapted grid (updating the modified adaptedgrid),

(6) If j > Jmin, set j = j − 1 and then following steps 2 through 6,(7) If j = Jmin, consider only steps 2 and 3. These points are added to the updated adapted grid

points, as the final stage.

The already-mentioned post processing procedure is illustrated in Fig. 2; there, distribution ofadapted grid points is shown in different levels of resolution. Solid points correspond to assumed


Figure 2. Modification of an adapted grid; solid points and hollow circles correspondto adapted and extra added points, respectively. For grid modification stage, we as-sume: Ns = Nc = 1.

s1

s2

s3

Figure 3. Adding procedure in the same resolution for point types s1, s2 or s3,where • ∈ V2, • ∪ ∈ V3, ∈ W2 and bright gray solid points are the added pointsbelonging to W2.

(original) adapted points and the hollow ones associate with points obtained after the post-processingprocedure. This modification leads to locally semi-uniform distributions.

3.2. 2-D grid modification. The 2-D grid modification will be done by controlling and adding newpoints in the same and coarser resolution levels, as the 1-D grid case. For this purpose, for differentpoints s1, s2 or s3 (Fig. 1), different adding procedures will be considered.

3.2.1. Adding extra new points by the multiresolution concept.Adding in the same resolution. Depending on point type (s1, s2 or s3), different inserting procedureswill be considered; different neighbor points will be added for each point s1, s2 or s3, see Fig. 3. There,for an adapted point belonging to W2 , new neighbor points of W2 are locally added. In this figure,bright gray solid points are the new extra points added around each point s1, s2 or s3. In this figure,only one row or column of the nearest points is considered for modification stage. In general, moresurrounding points of W2 can be considered for each point s1, s2 or s3.Adding in the successive coarser resolution. In this case, for a point belonging to Wj some newsurrounding points including in Wj−1 are added. To guarantee symmetry and gradual concentration ofmodified adapted points, it may also be necessary to consider some extra points from sub-space Vj−1.For different points s1, s2 and s3, different neighbor points will be considered. Such grid checking/adding


s1

w v w

w w w

s2

w

w

w

w

v

w

s3

w

w

v

w

Figure 4. Adding procedure in the successive coarser resolution for point types s1, s2or s3, where • ∈ V2, • ∪ ∈ V3, ∈ W2 and bright gray solid points are the addedpoints belonging to V2.

procedures are shown in Fig. 4. In each illustration, points denoted by w belong to sub-space Wj−1 =W2−1 and points with name v belongs to the sub-space Vj−1 = V2−1. In these figures, only one row orcolumn of the nearest distance is considered.

3.3. Post-processing 2-D adapted grids. The modification algorithm for 2-D grids is generallysimilar to the 1-D case; it can be reviewed as follows:

(1) Set the level resolution j (where Jmin ≤ j ≤ Jmax − 1) equal to the finest resolution level;i.e., j = Jmax − 1 (with spatial sampling steps dx = dy = 1/2Jmax−1),

(2) Consider the set of points corresponding to detail sub-space of resolution j; i.e., points (xj,k, yj,l) ∈Wj,(3) Control Ns neighbor-rows or columns of surrounding points for each side of the point (xj,k, yj,l);

this control is done at the same level of resolution,(4) Control Nc neighbor-rows or columns of points for each side of the point (xj,k, yj,l); this is done

at the successive coarser resolution (subspace Wj−1),(5) Regarding step 3, add the surrounding points in subspace Wj ,(6) Considering step 4, add the surrounding points in subspace Wj−1; in this stage, it may also be

needed to add some new points in the sub-space Vj−1 (as explained before),(7) If j > Jmin, set j = j − 1 and go back to step 2, else go to step 8,(8) If j = Jmin, consider only steps 2, 3 and 5.

4. Central and upwind-central high resolution schemes of spatially second orderaccuracy

Here, the semi-discretized form of a scalar first-order hyperbolic system, ut + F (u)x = 0, will beprovided for the KT method [32] and two other improvements of this scheme (to have less numericaldissipation) [30, 29]. By following the concept of the Reconstruction/Evolution/Projection (REP)procedure [32] (see Figure 5) for cell-centered non-uniform cells, it is easy to show that for jth cell, thesemi-discrete form is:

dujdt

+F ∗j+1/2 − F ∗

j−1/2

∆xj= 0, F ∗

j±1/2 := F(u∗j±1/2

),(4.1)

where uj denotes cell center solution (estimated state values) on jth cell; ∆xj := xj+1/2 − xj−1/2 is

the jth cell length; and F (u∗j±1/2) shows a proper combination of left and right reconstructed state values

and corresponding fluxes at cell edges xj±1/2. The left and right reconstructed state values are shown

respectively by uLj+1/2 and uRj+1/2. These values can be evaluated as uLj+1/2 := ui +(ux)j(xj+1/2 − xj

)and uRj+1/2 := uj+1 − (ux)j+1

(xj+1 − xj+1/2

)for spatially second order methods, where (ux)j denotes

a limited slope at point xj .


xi-1 xi- 12

xi xi+ 12

xi+1

ui-1n

uin

ui+1n

wi+ 12

n+1

wi- 12

n+1

win+1

wi+1n+1

wi-1n+1

uin+1

xi- 12 ,r xi+ 1

2 ,l xi+ 12 ,r xi+ 3

2 ,lxi- 12 ,lxi- 3

2 ,r xi- 12 ,r

Reconstruction

Evolution

Projection

Figure 5. The reconstruction/evolution/projection concept to have a second orderaccuracy in the spatial domain.

The two improvements of the central KT scheme are central-upwind methods, noting here by M1and M2. They use two different maximum local propagation speeds for right and left directions at celledges xj+1/2. The speeds are shown by aRj+1/2 and aLj+1/2 for right and left directions, respectively.

Such speed distinguishing leads to a narrower non-smooth zone around cell edge xj+1/2, and thereforeless dissipative schemes. The scheme M2 is an improvement of the M1 method using a narrowernon-smooth zone; for more details see [30, 29].

Expressed in terms of the left and right states FL/Ri+1/2 := F (u

L/Ri+1/2), the reconstructed fluxes for the

KT, M1 and M2 schemes are:

(1) The central KT scheme:

F ∗i+1/2 :=

[FRi+1/2 + FL

i+1/2

]− ai+1/2

[uRi+1/2 − uLi+1/2

].(4.2)

(2) For the central-upwind method M1:

F ∗i+1/2 :=

aRi+1/2FLi+1/2 − aLi+1/2F

Ri+1/2

aRi+1/2 − aLi+1/2

+ aLi+1/2aRi+1/2

uRi+1/2 − uLi+1/2

aRi+1/2 − aLi+1/2

.(4.3)

(3) For the central-upwind method M2:

F ∗i+1/2 :=

aRi+1/2FLi+1/2 − aLi+1/2F

Ri+1/2

aRi+1/2 − aLi+1/2

+aLi+1/2a

Ri+1/2

2×uRi+1/2 − uLi+1/2

aRi+1/2 − aLi+1/2

.(4.4)

5. Central high-resolution schemes on non-centered non-uniform 1-D cells

By using the cell-centered non-uniform cells, the TVD stability condition does not satisfy withoutaltering definition of slope/flux limiters. To preserve the TVD condition without limiter definitionmodification, it will be shown that by shifting cell centers in some cells, the TVD condition can bereached. These cells act as transmitting cells connecting two surrounding uniform cells of differentlengths. By such cell-center shifting, a new source of the truncation error will be introduced. Effects ofthis new error will be discussed.

Considering the REP, the full-discrete and semi-discrete forms of the scaler hyperbolic equation ut+F (u)x = 0 will be provided on non-centered non-uniform cells. For generality, it is assumed thatcell centers xj are not located at cell centers; so, the left and right cell edge positions are: xj−1/2 :=xj + (1 − pj)∆xj and xj+1/2 := xj + pj∆xj , where 0 < pj < 1 (for pj = 1/2 the cell center xj is the

middle point of jth cell).


5.1. The reconstruction and evolution stages.

(1) The spatio-temporal volume ∆xj+1/2×∆t is considered, where ∆xj+1/2 := xnj+1/2,r−xnj+1/2,l =

2∆tanj+1/2, and ∆t := tn+1 − tn. By averaging on this volume and considering the midpoint

integration rule in time, the solution wn+1j+1/2 := w(xj+1/2, t

n) can be obtained as:

wn+1j+1/2 =

1

∆xj+1/2

∫ xnj+1/2,r

xnj+1/2,l

u(x, tn+1) dx

=1

∆xj+1/2

∫ xnj+1/2,r

xnj+1/2,l

u(x, tn) dx− 1

∆xj+1/2

∫ tn+1

tn

[F (un

j+1/2,r)− F (unj+1/2,l)

]dt

=1

4

(−∆tan

j+ 12(ux)

nj +∆tan

j+ 12(ux)

nj+1 + 2∆xjpj (ux)

nj + 2∆xj+1 (pj+1 − 1) (ux)

nj+1 + 2uj + 2uj+1

)− 1

2anj+ 1

2

(F

n+ 12

j+ 12,r− F

n+ 12

j+ 12,l

),

(5.1)

where Fn+ 1

2

j+ 12 ,r

:= F(u(xj+ 1

2 ,r, tn+

12

))and F

n+ 12

j+ 12 ,l

:= F(u(xj+ 1

2 ,l, tn+

12

)).

(2) The spatio-temporal volume ∆xj ×∆t is considered, where ∆xj := xnj+1/2,l−xnj−1/2,r = ∆xj −

∆t(anj−1/2 + anj+1/2

). By averaging on this volume and considering the midpoint rule in time,

evolved solution wn+1j := w(xj , t

n+1) is:

wn+1j =

1

∆xj

∫ xnj+1/2,l

xnj−1/2,r

u(x, tn+1) dx

=1

∆xj

∫ xnj+1/2,l

xnj−1/2,r

u(x, tn) dx− 1

∆xj

∫ tn+1

tn

[F (un

j+1/2,l)− F (unj−1/2,r)

]dt

=1

2

(∆t

(anj− 1

2− an

j+ 12

)(ux)

nj +∆xj (2pj − 1) (ux)

nj + 2uj

)−

∆t

(F

n+ 12

j+ 12,l− F

n+ 12

j− 12,r

)∆xj −∆t

(anj− 1

2

+ anj+ 1

2

) .(5.2)

(3) Regarding volume ∆xj−1/2×∆t, where ∆xj−1/2 := xnj−1/2,l−xnj−1/2,r, evolved solution wn+1

j−1/2 :=

w(xj−1/2, tn+1) can be obtained with the similar procedure explained in (5.1).

Considering wn+1j+1/2, w

n+1j , and wn+1

j−1/2 definitions in Eqs. (5.1) and (5.2), a piecewise-linear approx-

imation on the staggered grid in the evolution stage is (see Figure (5)):

w(x, tn+1) :=∑j

wn+1

j+1/2 + (ux)n+1j+1/2

(x− xj+1/2

)1[

xnj+1/2,l

,xnj+1/2,r

] + wn+1j 1[

xnj−1/2,r

,xnj+1/2,l

] ,

where (ux)n+1j±1/2 denote limited derivatives and 1[a,b] shows a unit function on spatial interval [a, b].

5.2. The projection stage. The fully discrete second-order central scheme can be obtained by aver-aging w(x, tn+1) on interval

[xj−1/2, xj+1/2

], as:

un+1j =

1

∆xj

∫ xj+1/2

xj−1/2

w(x, tn+1) dx

=1

4∆xj

[2

(∆t

(F

n+ 12

j− 12,l− F

n+ 12

j+ 12,l+ F

n+ 12

j− 12,r− F

n+ 12

j+ 12,r

)+∆xj

(∆xj (2pj − 1) (ux)

nj + 2uj

))+ 2∆tan

j− 12

(∆xj−1pj−1 (ux)

nj−1 −∆xj (pj − 1) (ux)

nj + uj−1 − uj

)+ 2∆tan

j+ 12

(∆xj (−pj) (ux)

nj +∆xj+1 (pj+1 − 1) (ux)

nj+1 − uj + uj+1

)−∆t2

(anj− 1

2

)2 ((ux)

nj−1 + (ux)

nj − 2 (ux)

n+1j−1/2

)+∆t2

(anj+ 1

2

)2 ((ux)

nj + (ux)

nj+1 − 2 (ux)

n+1j+1/2

)].

(5.3)


5.3. The semi-discrete form. Regarding the full discrete form ((5.3)), for small enough ∆t values,

the following approximations can be assumed: 1) ignoring of terms corresponding to ∆t2; 2) Fn+ 1

2

j± 12 ,r

≈

Fnj± 1

2 ,r= FR

j±1/2 and Fn+ 1

2

j± 12 ,l

≈ Fnj± 1

2 ,l= FL

j±1/2 (due to the Taylor expansion in smooth zones). By

these approximations, Eq. (5.3) can be rearranged and rewritten as:

un+1j − un

j

∆t=

∆uj

∆t≈ 1

4∆xj

2(−Fn

j+ 12,l − Fn

j+ 12,r + Fn

j− 12,l + Fn

j− 12,r

)− 2

(−uj−1a

nj− 1

2+ uja

nj− 1

2+ uja

nj+ 1

2− uj+1a

nj+ 1

2

)+ 2

(∆xj−1pj−1a

nj− 1

2(ux)

nj−1 +∆xj

(anj− 1

2− pj

(anj− 1

2+ an

j+ 12

))(ux)

nj

)+2

(∆xj+1 (pj+1 − 1) an

j+ 12(ux)

nj+1

)−

(1

2− pj

)(ux)

nj

∆xj

∆t.

(5.4)

Since uLj+1/2 := uj + pj∆xj (ux)nj and uRj−1/2 := uj − (1− pj)∆xj (ux)

nj , (5.4) can be rewritten as:

∆uj∆t

+F ∗j+1/2 − F ∗

j−1/2

∆t≈ Q,(5.5)

where F ∗j+1/2 :=

FRj+1/2+FL

j+1/2

2 − aj+1/2

2

(uRj+1/2 − uLj+1/2

)and Q := −

(12 − pj

)(ux)

nj

∆xj

∆t .

We conclude this section with the following remarks regarding ((5.5)).

(1) The term Q acts as a new source of the truncation error,(2) Effects of ∆t in Q can be controlled by small-enough spatial sampling steps ∆xj . This can be

done by a proper grid adaptation procedure around high-gradient zones,(3) If xj is in the middle of the jth cell, i.e. p = 1/2, the truncation error Q vanishes: Q = 0,(4) Around discontinuities, since limited slopes (ux)

nj are nearly zero, the error Q approaches to

zero,(5) Average solution on jth cell is

(unj)ave

= unj +Q×∆t.

6. Numerical entropy production of central/central-upwind schemes

Calculation of numerical entropy production helps one to control quality/uniqueness of numericalresults especially for ones without exact solutions. This is because, theoretically, the numerical entropyproduction is zero in smooth regions while less than zero around shocks and discontinuities.

Puppo [43, 44] showed how to estimate the numerical entropy production for staggered central highresolution schemes. After these works, Puppo et al. [45] improved the previous works for non-staggeredcentral/central-upwind schemes.

The entropy function η(u) with flux ψ(u) satisfies the relationship:

ηt + ψx(u) ≤ 0, η := η(x).

Following works [43, 44], by integrating this relationship in spatio-temporal volume[xj − ∆xj

2 , xj +∆xj

2

]×[

tn, tn+1], and discretizing the inequality by considering the REP concept (used in the KT, M1 or M2

method), we have:

ηn+1j −

(ηnj − 1

∆xj

∫ tn+1

tn

[ψ∗j+1/2 − ψ∗

j−1/2

])≤ 0, ηnj := η(xj , t

n).(6.1)

Based on this inequality, the density of numerical entropy production at xj , Snj , can be defined as:

Snj =

1

∆xj

ηn+1j −

(ηnj − 1

∆xj

∫ tn+1

tn

[ψ∗j+1/2 − ψ∗

j−1/2

]), Sn

j := S(xj , tn),(6.2)


where ψ∗j±1/2 := ψ∗

j±1/2

((uLj±1/2

)n,(uRj±1/2

)n).

In Eqs. (6.1) and (6.2), the terms in parentheses estimate the evolved values of the entropy at thenext time step (this can be done with the KT, M1 or M2 method). Theoretically, as mentioned before,the parameter Sn

j is zero in smooth regions, however, in numerical solutions, it may be slightly lessor more than zero [45]. This kind of entropy definition cannot efficiently detect some phenomena likecontact discontinuities even by increasing number of resolution levels (see the Sod and the Lax problemsin the Euler gas dynamic system). To remedy this, other approaches like the entropy viscosity schemecan be recommended [21].

7. Local truncation error on non-uniform cells

7.1. One dimensional systems. The aim of this section is to estimate local truncation errors inhyperbolic systems of conservation laws with the governing equation: ut + F (u)x = 0, u(x, t = 0) =u0(x).

It is easy to show that solution of this equation is also a weak solution satisfying the relationship:

E(u, ϕ) =−∫ ∞

t=0

∫X

u(x, t)ϕt(x, t) + F (u(x, t))ϕx(x, t)dxdt

+

∫X

u(x, t)ϕt(x, t)dx = 0,

(7.1)

where ϕ (x, t) is a test function that ϕ (x, t) ∈ C∞0 (X × [0,∞])). One effective and practical way to

measure convergence of a numerical solution u(x, t) is to check how much it fails to satisfy (7.1); this canbe measured by evaluating E (u, ϕ) [28, 27]. For a convex scalar hyperbolic system (dF (u)/d(u) > α >

0), the function E (u, ϕ) measures point-wisely real errors in weak Lip′-norm; this norm is developed

by Tadmor to measure errors in simulation of the non-linear scalar conservation laws [52, 53]. Innonlinear conservation laws, discontinuous solutions develop typically; in these cases standard methodsof error estimation are not valid. Such approaches consider the Taylor expansion which is based on thesmoothness assumption.

The final point is the relationship of this local truncation error with the weak Lip′-norm theory

studied for convex scalar one dimensional hyperbolic systems. Numerical results confirm that it is alsoan effective tool for systems of one dimensional PDEs, 2-D problems, and even non-convex systems.

To be sure that E (u, ϕ) measures errors of u(x, t), some constraints should be met by the testfunction ϕ(x, t): 1) the space spanned by the test functions should have a higher order accuracy, sothat the order of error E (u, ϕ) is affected by solutions u(x, t); 2) the test function have continuousderivatives; 3) to estimate locally the error E (u, ϕ), the test function has to be compact support inspatio-temporal domains.

We now consider the truncation error Enj := E(u∆, ϕ) for the piecewise constant approximate solution

u∆(x, t)

u∆(x, t) =∑j,n

unj 1Cj×Tn(x, t), Cj × Tn :=[xj−1/2, xj+1/2

]×[tn−1/2, tn+1/2

],

The compact test function ϕ(x, t) is assumed to be ϕnj (x, t) := Bj(x)Bn(t), where Bj(x) and B

n(t) arethe quadratic B-splines with center points x = xj and t = tn. In this case, supports of Bj(x) and B

n(t)

belong to x ∈ [xj−3/2, xj+3/2] and t ∈ [tn−3/2, tn+3/2], respectively.To obtain higher-order B-splines, the recurrence feature between higher-order and lower-order B-

splines: recall that the B-spline of order k with centered position j can be obtained as,

Bj,k = Bj,k−1wj,k +Bj+1,k (1− wj+1,k) , wj,k(x) :=

x−xj

xj+k−1−xj, if xj = xj+k−1,

0, otherwise.(7.2)

B-splines should also satisfy the partition of unity condition∑

j Bj,k(x) ≡ 1. For second order spline

(7.2) leads to:

Bj,2 = Bj,1wj,2 +Bj+1,2(1− wj+1,2),


Let us consider three successive cells, with length ratios, a = ∆xj−1/∆xj and b = ∆xj+1/∆xj ,where ∆xj := xj+1/2 − xj−1/2 denotes jth cell length with cell middle point xj . It is easy to showthat Bj(x) is:

Bj(x) =

(2x−2xj+∆x+2a∆x)2

4a(1+a)∆x2 , if xj−3/2 ≤ x ≤ xj−1/2,−4(2+a+b)(x−xj)

2−4(a−b)(x−xj)∆x+(2+3b+a(3+4b))∆x2

4(1+a)(1+b)∆x2 , if xj−1/2 ≤ x ≤ xj+1/2,(−2x+2xj+∆x+2b∆x)2

4b(1+b)∆x2 , if xj+1/2 ≤ x ≤ xj+3/2,

0, otherwise,

(7.3)

where: ∆x := ∆xj ; xj = (xj−1/2+xj+1/2)/2; xj−3/2 := xj−∆x(1/2+a); xj−1/2 := xj−∆x/2; xj+3/2 :=xj +∆x(1/2 + b); xj+1/2 := xj +∆x/2.

It is easy to check that on uniform grids, where a = b = 1, the above-mentioned B-spline definitionleads to the B-spline on uniform grids. Since in time-domain, in this study, a constant time-step isused, the definition of Bn(t) is the quadratic B-spline on uniform grids with step ∆t, as:

Bn(x) =

(t−tn−3/2)2

2∆t2, if tn−

32 ≤ t ≤ tn−

12 ,

34 − (t−tn)2

∆t2, if tn−

12 ≤ t ≤ tn+

12 ,

(t−tn+3/2)2

2∆t2, if tn+

12 ≤ t ≤ tn+

32 ,

0, otherwise,

(7.4)

here: tn±12 :=

(tn ± ∆t

2

)and tn±

32 :=

(tn ± 3

2∆t).

For spatially non-uniform grid points, it is straightforward to show that the local truncation errorcan be expressed as:

Enj = ∆xUn

j +∆tFnj ,(7.5)

where Unj is expressed in terms of the time differences ∆unα := 1/2

(un+1α − un−1

α

),

Unj =

a2

3(a+ 1)∆un

j−1 +(a(3b+ 2) + 2b+ 1)

3(a+ 1)(b+ 1)∆un

j +b2

3(b+ 1)∆un

j+1,

and Fnj is expressed in terms of the time averages µFn

α := 1/6(Fn−1α + 4Fn

α + Fn+1α

),

Fnj = − (b− a)

(a+ 1)(b+ 1)µFn

j − a

a+ 1µFn

j−1 +b

b+ 1µFn

j+1.

7.2. Two-dimensional problems. Let us consider a two dimensional scalar hyperbolic PDE as:

ut + F (u)x +G(u)y = 0, u0(x, y) := u(x, y, t = 0).(7.6)

As in the 1-D case, the weak solution of (7.6) satisfies

E(u, ϕ) =

−∫ ∞

t=0

∫Ω

u(x, y, t)ϕt(x, y, t) + F (u(x, y, t))ϕx(x, y, t) +G (u(x, y, t))ϕy(x, y, t) dxdydt

+

∫Ω

u(x, y, t)ϕt(x, y, t) dxdy = 0.

(7.7)

We quantify the truncation error Enj,k := E(u∆, ϕ) for the piecewise constant approximate solution,

u∆(x, y, t),

u∆(x, y, t) :=∑j,k,n

unj,k1Cj,k×Tn(x, y, t) Cj,k × Tn :=[xj−1/2, xj+1/2

]×[yk−1/2, yk+1/2

]×[tn−1/2, tn+1/2

].

The test function ϕ(x, y, t) can also be defined as: ϕnj,k(x, y, t) := Bj(x)Bk(y)Bn(t). Where all func-

tions Bj(x), Bk(y) (from (7.3)), and Bn(t) (from (7.4)) denote quadratic B-splines. Let us assume the

successive cell length ratios in x and y directions to be:a =

∆xj−1

∆xj, b =

∆xj+1

∆xj

and

c = ∆yk−1

∆yk, d = ∆yk+1

∆yk

,


where: ∆x = ∆xj := xj+1/2 − xj−1/2 and ∆y = ∆yk := yk+1/2 − yk−1/2. By these assumptions, thelocal truncation error En

j,k can be represented as:

Enj,k =

∆x∆yUn

j,k +∆t∆yFnj,k +∆t∆xGn

j,k

,(7.8)

where Unj,k is expressed in terms of the time differences ∆unα,β := 1/2

(un+1α,β − un−1

α,β

)Unj,k =

a2d3

9(a+ 1)b(b+ 1)∆un

j−1,k+1 +a2(a(3b+ 2) + 2b+ 1)un−1

j−1,k

9(a+ 1)2(b+ 1)∆un

j−1,k+

b2c3

9a(a+ 1)(b+ 1)∆un

j+1,k−1 +b2(a(3b+ 2) + 2b+ 1)

9(a+ 1)(b+ 1)2∆un

j+1,k+

c3(a(3b+ 2) + 2b+ 1)

9a(a+ 1)2(b+ 1)∆un

j,k−1 +d3(a(3b+ 2) + 2b+ 1)

9(a+ 1)b(b+ 1)2∆un

j,k+1+

(a(3b+ 2) + 2b+ 1)2

9(a+ 1)2(b+ 1)2∆un

j,k +ac3

9(a+ 1)2∆un

j−1,k−1 +bd3un−1

j+1,k+1

9(b+ 1)2∆un

j+1,k+1,

and Fnj,k and Gn

j,k are expressed in terms of the time averages µZnα,β := 1/6

(Zn−1

α,β + 4Znα,β + Zn+1

α,β

),

Fnj,k =− c3

3(a+ 1)2µFn

j−1,k−1 +(a− b)c3

3a(a+ 1)2(b+ 1)µFn

j,k−1 +bc3

3 (a2 + a) (b+ 1)µFn

j+1,k−1

− a(2b+ a(3b+ 2) + 1)

3(a+ 1)2(b+ 1)µFn

j−1,k − ad3

3(a+ 1) (b2 + b)µFn +

(a− b)(2b+ a(3b+ 2) + 1)

3(a+ 1)2(b+ 1)2µFn

j,k

(a− b)d3

3(a+ 1)b(b+ 1)2µFn

j,k+1 +b(2b+ a(3b+ 2) + 1)

3(a+ 1)(b+ 1)2µFn

j+1,k +d3

3(b+ 1)2µFn

j+1,k+1,

Gnj,k =− c2b2

3 (a2 + a) (b+ 1)µGn

j+1,k−1 +(a− b)b2

3(a+ 1)(b+ 1)2µGn

j+1,k +d2b

3(b+ 1)2µGn

j+1,k+1

− ac2

3(a+ 1)2µGn

j−1,k−1 +a2(a− b)

3(a+ 1)2(b+ 1)µGn

j−1,k +a2d2

3(a+ 1) (b2 + b)µGn

j−1,k+1

− c2(2b+ a(3b+ 2) + 1)

3a(a+ 1)2(b+ 1)µGn

j,k−1 +(a− b)(2b+ a(3b+ 2) + 1)

3(a+ 1)2(b+ 1)2µGn

j,k +d2(2b+ a(3b+ 2) + 1)

3(a+ 1)(b+ 1)2bµGn

j,k+1.

8. The TVD property of semi-discrete schemes

8.1. Global and local TVD conditions. To control the development of spurious oscillations innumerical simulation of hyperbolic systems, it is necessary to show that a monotone (non-increasing ornon-decreasing) profile remains monotone during time evolution. To achieve high-order of accuracy, arelaxed monotonicity condition — so called Total Variation Diminishing (TVD) is sought [22, 34]. Weconsider a general class of non-linear semi-discrete scalar scheme which takes the incremental form

duidt

= C+i+1/2(ui+1 − ui)− C−

i−1/2(ui − ui−1), ui := ui(t).

A general (global) TVD condition derived in [22] is expressed in terms of the global positivity condition,e.g.,[24, 41, 46]:

C+i+1/2 ≥ 0, C−

i+1/2 ≥ 0, C+i+1/2 + C−

i+1/2 ≤ 1.(8.1)

Local TVD conditions were studied in [51] where it was shown that for the TVD property to hold, itsuffice to verify the positivity condition (8.1) in the extreme cells and their surrounding grid points. Inthe present context of (4.1), these local TVD conditions amount to:

(1) In extreme points, for both uniform and non-uniform cells, the numerical fluxes F ∗i±1/2 satisfy

(see [51], Lemma 2.1):(a) at maximum values ui : F

∗i+1/2 ≥ F ∗

i−1/2,

(b) at minimum values ui : F∗i+1/2 ≤ F ∗

i−1/2,


(2) Oshor [42] showed that in order to satisfy the positivity conditions, and therefore the TVDproperty, we should have:0 ≤ ∆x

∆−ui(u,x)i ≤ 1 and 0 ≤ ∆x

∆+ui(u,x)i ≤ 1,

(3) For neighbor points of an extreme ui (with estimated derivative (u,x)i = 0; this is based oncondition 2), for uniform grids with width ∆x, the neighbor derivatives meet conditions (see[51], example 2.4):12 | ∆x

∆−ui(u,x)i−1 |≤ 1 and 1

2 | ∆x∆+ui

(u,x)i+1 |≤ 1,

In these conditions, we have: ui := u(xi); ∆xi := xi+1/2−xi−1/2 denotes ith cell width; in case of uniform

grids: ∆x = ∆xi; F∗i±1/2 represent reconstructed fluxes at right and left cell boundaries xi±1/2; ∆±ui :=

±(ui±1 − ui); (u,x)i±1 are the estimated (limited) first derivatives at points xi±1.The conditions 1-3 have physical meaning; the first condition states that an extreme solution with

maximum value should decrease in time (i.e., un+1i ≤ uni ), while an extreme solution with minimum

value should increase through time (i.e., un+1i ≥ uni ). From the second condition, it is clear that at

extreme points we must have: (u,x)i = 0. Due to the third condition, the reconstructed values incell edges should satisfy the monotonicity feature: magnitude of the reconstructed values should beless than immediately neighbor cell center values. This means, for example, if uni−1 ≤ uni ≤ uni+1,then uni−1 ≤ uni−1/2 ≤ uni ≤ uni+1/2 ≤ uni+1.

The third local TVD condition is for uniform cells; in case of non-uniform cells, due to the E-conditionat extreme values ui with estimated derivative (u,x)i = 0, we have:

Sign (ui+1 − ui) = Sign

((ui+1 −

∆xi+1

2(u,x)i+1

)− (ui)

),

Sign (ui − ui−1) = Sign

((ui)−

(ui−1 +

∆xi−1

2(u,x)i−1

)).

According to these relationships, TVD preserving derivatives should satisfy:

1

2| ∆xi−1

∆−ui(u,x)i−1 |≤ 1 and

1

2| ∆xi+1

∆+ui(u,x)i+1 |≤ 1

These two relationships confirm also as before necessity of the monotone reconstruction in cell edgesfor non-uniform cells.

8.2. Irregularity effects on slope limiters. The central/central-upwind schemes (e.g., the KTmethod) is originally developed for uniform cells. Most of them satisfy the TVD and monotonicitypreserving conditions. On irregular cells, these methods do not completely meet these conditions andtherefore do not remain necessarily stable. In the following, effects of cell irregularities will be studiedby considering the monotonicity preserving necessity and the local TVD conditions.

It will be shown that only the grid modification (for gradual variation of grids) is not generallyenough to guarantee TVD results.

8.2.1. Slope limiters on uniform grids. Most of slope limiters meet some necessary conditions, as: 1)the TVD; 2) preserving linear approximation; and 3) symmetric feature. These features are explained,in brief, as follows.The TVD property. Considering physical meanings of the local TVD conditions 1-3, the upper limitof a TVD limiter for ith cell is (for more details, one can see [4]):

u′i = min

2 (∆+ui)

∆x,2 (∆−ui)

∆x

,(8.2)

where ui denotes cell center solution on the ith cell and operator u′i is upper limit of a discrete approx-imation of grad(ui) in a way that solutions remain TVD. To have TVD solutions, estimated deriva-tive (ux)i := du(xi, t)/dx should be limited by a slope limiter ϕi := ϕ(Ri) as ϕi(ux)i. The function ϕiis a slope limiter at xi, where 0 ≤ ϕi ≤ 1. Parameter Ri measures smoothness by relative of successive

gradients around point xi; for uniform grids its definition is: Ri =∆+ui/∆+xi

∆−ui/∆−xi= ∆+ui

∆−ui.

For all TVD slope limiters, it is necessary that: ϕi(ux)i ≤ u′i (this upper TVD limiter will be derivedon non-uniform cells).


Linear approximation preserving. For a linear function with slope s on a uniform grid, it isclear: Ri = ∆+ui/∆−ui = s. (∆+xi) / s. (∆−xi) = 1. In this regard, the linear preserving conditionis: ϕ(1) = 1. This constraint can also be obtained by considering the MINMOD limiter definition: thelinear preserving feature is conserved if the forward and backward derivatives are the same, i.e., Ri = 1.The symmetric feature. The condition is: ϕ(Ri) = ϕ(1/Ri). This condition assure that slope limitereffects are the same for forward and backward propagating solutions. To more clarify this feature, a newparameter f is defined as: f = (∆−ui)/(∆0ui), where ∆0ui := ui+1 − ui−1. For monotone solutions, uiis always between ui−1 and ui+1, so f belongs always to the range [0, 1]. The parameters R and fhave then a relationship with each other, as: R = (1 − f)/f . The symmetric property is now clear,since: ϕ(f) = ϕ(R) = ϕ((1 − f)/f) and ϕ(1/R) = ϕ(f/(1 − f)) = ϕ(1 − f), then ϕ(f) = ϕ(1 − f)for 0 ≤ f ≤ 1.

8.3. Slope limiters on cell-centered non-uniform grids. A grid with non-uniform cells is con-sidered. Widths of successive cells can be related to each other by coefficients: a = ∆xi−1/∆xiand b = ∆xi+1/∆xi. Possible jump in solutions between points xj+1 and xj−1 is denoted by ∆0ui,where ∆0ui := ui+1 − ui−1. The forward (D+ui), backward (D−ui) and centered (D0ui) differences

can then be written as: D±ui :=∆±ui

∆±xi, D0ui :=

∆0ui

∆−xi+∆+xi. Considering definition of f , these deriva-

tives can be rewritten as: D+ui = (2(1 − f)∆0ui)/((1 + b)∆xi); D−ui = (2f∆0ui)/((1 + a)∆xi);and D0ui = (2∆0ui)/((2 + a+ b)∆xi).The TVD property. This can be obtained by the local TVD conditions 1-3 (see [4]). It has the samedefinition as the uniform case (Eq. (8.2)):

u′i = min

2∆+ui∆xi

,2∆−ui∆xi

.(8.3)

Linear approximation preserving. For non-uniform grids, the linear preserving condition for a slopelimiter is [4]:

ϕiDi (f = fp) =2∆0ui

(a+ b+ 2)∆xi,

where fp = a+1a+b+2 . The linear preserving feature is conserved if the backward and forward derivatives

are equal to each other: D+ui = D−ui. By considering the MINMOD definition, this happens forcase f = fp, or values of ϕiDi(fp) are values of the MINMOD limiter at the fp.Symmetric condition. This feature is not fulfilled, in general.

8.4. Performance of the generalized MINMOD limiter on non-uniform grids. The generalizedMINMOD (GMINMOD), can be rewritten as:

(ϕiDi)GMINMOD = GMINMOD θD−ui, D0ui, θD+ui , 1 ≤ θ ≤ 2.(8.4)

In case of non-uniform grids, different definitions of D0ui can be considered. One is based on theabove-mentioned definition: D0ui = ∆0ui/(∆−xi +∆+xi), and the other one obtains by the first orderleast square based estimated slope, as [4]:

D0ui = DLsq.ui :=∆+x

2i

∆−x2i +∆+x2iD+ui +

∆−x2i

∆−x2i +∆+x2iD−ui.(8.5)

On uniform grids, both definitions lead to the first order central difference equation: (ui+1−ui−1)/(2∆xi).Performance of these two GMINMOD limiters will be illustrated on non-uniform grids.

Regarding f and ∆0uj definitions, functions DLsq.ui (Eq. (8.5)) and u′i (Eq. (8.3)) can be rewritten

as:

D0ui = DLsq.ui =(1 + b)(1− f) + (1 + a)

(1 + b)2 + (1 + a)2

f.∆0ui∆xi

,

u′i = minf, 1− f2∆0ui∆xi

.


xj-1 xj xj+1

xj-32 xj-12 xj+12 xj+32

xj-1 xj xj+1

xj-32 xj-12 xj+12xj+32

xj-1 xj xj+1

xj-32 xj-12 xj+12 xj+32

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

ΦiH

u xL i=HD

0uiD

x iL´

HaLa=1b=1

linearitypreserving

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

fΦ

iHu xL i=HD

0uiD

x iL´

HbLa=2.00b=0.5

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

ΦiH

u xL i=HD

0uiD

x iL´

HcLa=1.33b=0.667

u ¢iGMINMOD, Θ=1

GMINMOD, Θ=2; D0ui=D0ui D0xi

GMINMOD, Θ=2; D0ui=DLsq.ui

Figure 6. Performance of GMINMOD limiters on non-uniform grids. Regarding thetop row in each figure, hollow circles and solid points represent center of cells and celledges, respectively.

Performance of the GMINMOD limiters on uniform and non-uniform grids are shown in Figure 6.Figure 6(a) is for uniform case and Figures 6(b,c) are for non-uniform cases. In all figures, center ofeach cell, xi, is in the middle of ith cell, i.e.:xi = (xi+1/2 + xi−1/2)/2. A gradual variation of celllengths is considered for irregular grids. In Figure 6(b) length of cells are: ∆xi−1 = 2dx, ∆xi = dx,and ∆xi+1 = 0.5dx; and in Figure 6(c) the lengths are: ∆xi−1 = 2dx, ∆xi = 1.5dx, and ∆xi+1 = dx.The results confirm that: 1) on gradually varying grids, limiters may not completely remain in theTVD region; 2) the symmetry condition may not satisfy; 3) more gradual variation of grids is, morestability exists; 4) the long-term numerical stability cannot guarantee; and 5) the GMINMOD limiterby the direct-definition of central differencing, D0ui = (∆0ui)/(∆−xi+∆+xi), leads to more symmetricbehaviors, so this definition will be considered in this work.

It will be shown that to have a TVD solution (defined on non-uniform grids) without modifying limiterdefinitions (the GMINMOD limiter, here), the cell middle point, xi, should be shifted slightly; and thisis only necessary for transmitting cells (a cell between two surrounding uniform cells with different celllengths). In the following, at first, it will be shown how to choose properly cell centers/edges by theadaptive wavelet transform. It will then mathematically be proved that why such spatial configurationslead to stable and TVD solutions without modifying limiter definitions.

8.5. Choosing of cell centers and edges by the wavelet-based adaptation algorithm. In thiswork, at first, cell center positions, xj , are evaluated by the adaptive wavelet transform, and then, celledges are simply assumed to be the middle point of them, as: xj+1/2 = (xj + xj+1)/2.

The interpolating wavelet theory in Sec. 2 uses the pyramid algorithm. In this formulation, distancebetween detail coefficients in the resolution j is twice those in the resolution j+1. Consider an adaptedgrid where for every adapted points of resolution level j+1, there exist always two surrounding adaptedpoints of resolution level j. For such adapted grids, inter-distances of successive points increase or de-crease gradually by the dyadic pattern. Let us consider an adapted cell-centers as · · · , 2d, 2d, d, d, · · · .This means, for grid points · · · , xj−2, xj−1, xj , xj+1, xj+2, · · · , we have: xj−1−xj−2 = xj−xj−1 = 2dxand xj+1 − xj = xj+2 − xj+1 = dx. Such configuration is illustrated in Figure 7(a). For this adaptedpoints, it is assumed that there exists always three successive points with equal distances from eachother. In this figure, cell edges are middle point of two successive cell centers, i.e.: xj+1/2 = (xj+xj+1)/2.

By this, length of created cells are: · · · ,∆xj−1 = 2dx,∆xj = 1.5dx,∆xj+1 = dx, · · · . Note that jth

cell acts as a transiting cell with a shifted cell center (xj is no longer in the middle of cell j). In all thesurrounding cells, all cell centers remain in the middle of cells. In the next subsection, it will be provedthat such cell configuration leads to stable and TVD results. This cell configuration can obtain by thepost-processing stage, Sec. 3.


HaL

x j-2 x j-1 x j x j+1 x j+2

x j-32 x j-12 x j+12 x j+32

D-x j-1=2 dx D-x j=2 dx D-x j+1=dx D+x j+1=dx

Dx j-1=2dx Dx j=1.5dx Dx j+1=dx

H1-pLDx j p Dx j

xj-1 xj xj+1

xj-32 xj-12 xj+12 xj+32

xj-2 xj-1 xj

xj-52 xj-32 xj-12 xj+12

xj xj+1 xj+2

xj-12 xj+12 xj+32 xj+52

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

ΦiH

u xL i=HD

0uiD

x iL´

HbLa=43b=23p=13

linearitypreserving

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

ΦiH

u xL i=HD

0uiD

x iL´

HcLa=1b=34p=13

linearitypreserving

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

ΦiH

u xL i=HD

0uiD

x iL´

HdLa=32b=1p=13

linearitypreserving

u ¢i GMINMOD, Θ=1 GMINMOD, Θ=2; D0ui=D0ui D0xi

Figure 7. Wavelet based adaptive distribution of cell centers xj , corresponding edgelocations and TVD feature of the GMINMOD limiter on resulted non-uniform grids.

8.6. Stability and TVD conditions on wavelet-based adapted grids. In the proposed wavelet-based grid adaptation, as mentioned, cell center of transmitting cells do not remain in the middle ofcells, Figure 7(a). In Figure 7, three pattern of successive cells are distinguishable: 1) j − 2, j − 1, j;2) j − 1, j, j + 1; and 3) j, j + 1, j + 2. These sets are shown in Figures 7 (b) to (d). In the set (2),the transmitting cell j is in the middle, while in the remaining groups, the transmitting cell j is thefirst or the last cell.

In the following, the TVD local conditions are checked and modified for such spatially non-centeredcell centers. This will be done for the set (2) (see Figure 7(b)), and then it will be checked for thegroups (1) and (3).

8.6.1. The TVD condition when transmitting cell is in the middle of surrounding cells (the set (2)).To provide the TVD condition for the cell set j − 1, j, j + 1 (see Figure 7(b)), a right propagatingscalar advection equation is considered as: ut + aux = 0 for a > 0. For simplicity, the forward Eulerdiscretization in time will be used. The upwind finite volume method with second order accuracy willbe considered for the spatial discretization. The resulted discretized system is:

un+1j = unj − a∆t

∆xj

(uLj+ 1

2− uLj− 1

2

).(8.6)

The symbol L represents the upwind flux and therefore the upwind-based reconstructed valueof uLj+1/2 is:

uLj+ 12= uj + (p∆xj)Sj ,

where Sj := ϕjDj is a limited slope at point xj . Let assume in the transmitting cell j, the cell center xjlocates in a way that xj+1/2−xj = p∆xj and xj −xj−1/2 = (1−p)∆xj . For the wavelet based adaptedgrids, it is easy to show that p = 1/3.


To show a method is TVD, it should be confirmed that: 1) a monotone increasing (or decreasing)solution remains monotone increasing (or decreasing) in time (the monotonicity preserving feature:resulted from the posivity condition [51]); 2) if unj is a local maximum (or minimum), then at the next

time step: un+1j ≤ unj (or un+1

j ≥ unj ) (due to the first local TVD condition).Controlling of the monotonicity preserving condition. Assume a monotonically increasing so-lution at time step t = tn as: unj−1 ≤ unj ≤ unj+1. This feature should satisfy at the next time step,

i.e.: un+1j−1 ≤ un+1

j ≤ un+1j+1 .

For the monotone increasing solutionunj, slopes Sj−1 and Sj are positive, so: unj−1/2 = unj−1 +

(∆xj−1/2)Sj−1 ≤ unj and unj+1/2 = unj + (p∆xj)Sj ≥ unj . These inequalities are obtained due to the

monotone reconstruction constraint: the local TVD condition 3. Hence from Eq. (8.6), we have:

un+1j ≤ unj − a∆t

∆xj

(unj − unj

)= unj .(8.7)

To complete the proof, we need to estimate a below bound for un+1j . At first, due to the local TVD

constraint on reconstruction edge values (the condition 3), it is clear that: unj − (1− p)∆xjSj ≥ unj−1.By considering this relationship, and conditions Sj ≥ 0 & Sj−1 ≥ 0, from Eq. (8.6), we have:

un+1j = unj − λj

((unj + (p∆xj)Sj

)−(unj−1 +

(∆xj−1

2

)Sj−1

))≥ unj − λj

((unj +

[unj − unj−1 −∆xjSj

])−(unj−1

))≥ unj − λj

((unj + unj − unj−1

)− (unj−1)

)= unj (1− 2λj) + 2λju

nj−1,

so:

un+1j ≥ unj−1,(8.8)

where λj := a∆t/∆xj . The relationship ((8.8)) valids for λj ≤ 0.5. Eqs. (8.7) and (8.8) lead to

the condition un+1j−1 ≤ unj−1 ≤ un+1

j ≤ unj ≤ · · · . This means a monotone solution remains monotonethrough time.Controlling of the extreme conditions (the first local TVD condition).

(1) local maximum: let have a right propagating wave, a > 0 and unj+1 is a local maximum ongroup cell (2) (Figure 7(b)). Due to the extreme condition, we have: 1) Sj+1 = 0 (the secondlocal TVD condition); 2) unj+1 ≥ unj and unj+1 ≥ unj−1; and 3) Sj ≥ 0 (due to the monotonereconstruction feature). Rewriting Eq. (8.6) for the maximum point, we have:

un+1j+1 = unj+1 −

a∆t

∆xj+1

(uLj+ 3

2− uLj+ 1

2

)= unj+1 −

a∆t

∆xj+1

([unj+1 + 0.5∆xi+1Sj+1

]−[unj + p∆xjSj

])= unj+1 −

a∆t

∆xj+1

([unj+1

]−[unj + p∆xjSj

]).

To guarantee that un+1j+1 ≤ unj+1, the term in the parentheses should be non-negative; hence:

0 ≤ Sj ≤unj+1 − unjp∆xj

=∆+u

nj

p∆xj.(8.9)

(2) local minimum let the wave propagate to the left, a < 0 and unj−1 is a local minimum. Thismeans: unj−1 ≤ unj , u

nj−1 ≤ unj−2, Sj−1 = 0 and Sj ≥ 0. By the spatio-temporal discretization


((8.6)) at xj−1, we have:

un+1j−1 = unj−1 −

a∆t

∆xj−1

(uLj− 1

2− uLj− 3

2

)= unj−1 −

a∆t

∆xj−1

([unj + (1− p)∆xjSj

]−[unj−1 + 0.5∆xj−1Sj−1

])= unj−1 −

a∆t

∆xj−1


]−[unj−1

]).

Since a < 0 and due to the constraint un+1j−1 ≥ unj−1, we should have:


]−[unj−1

])≥

0. This means:

0 ≤ Sj ≤unj − unj−1

(1− p)∆xj=

∆−unj

(1− p)∆xj.(8.10)

From Eqs. (8.9) and (8.10), the TVD constraint on non-uniform grids is:

Sj = u′j = min

∆−u

nj

(1− p)∆xj,∆+u

nj

p∆xj

.(8.11)

8.6.2. The TVD condition when transmitting cell is the first or last cell (the set (1) or (3)). In thiscase, it is easy to show that the TVD limiter constraint, u′j is the same as cell-centered cells, see Eq.(8.3).

8.7. Controlling of the TVD condition for slope limiters on wavelet-based adapted grids.A modified wavelet-based adapted grid is considered; the modification is done with the post-processingstage. It is assumed also there exists always at least three neighbor cell centers of equal distances fromeach other, as explained in Sec. 8.5 and illustrated in Figure 7(a). In this figure, we have: xj−1−xj−2 =2dx, xj − xj−1 = 2dx, xj+1 − xj = dx and xj+2 − xj+1 = dx. This configuration of grid points leadsto stable and TVD solutions, which will be studied later. Depending of the transmitting cell location,three cell sequences are detectable: 1) j − 2, j − 1, j; 2) j − 1, j, j + 1; 3) j, j + 1, j + 2 (Figure7(a)).Cell sequence j − 1, j, j + 1: Transmitting cell j (with a shifted cell center) is the middleone. In this case, we have ∆−xj = [a/2 + (1 − p)]∆xj and ∆+xj = [p + b/2]∆xj . So, the backward,forward and centeral derivatives (in the GMINMOD limiter, Eq. (8.4)) can be rewritten as: D−uj =

2f∆0uj

[a+2(1−p)]∆xj, D+uj =

2(1−f)∆0uj

[2p+b]∆xj, and D0uj =

2∆0uj

(2+a+b)∆xj. Where a := ∆xj−1/∆xj = 4/3 & b :=

∆xj+1/∆xj = 2/3; p measures cell-center shifting and here p = 1/3 (see Figures 7(a)-(b)).The functions (ϕj)TVDDj (Eq. (8.11)) and GMINMOD (Eq. (8.4)) are illustrated in Figure 7(b).

The comparison offers: 1) the limiter remains completely in the TVD region; 2) the linear preservingfeature is satisfied; 3) at the expense of the symmetric feature, the transmitting cell j acts properly forjoining surrounding cells; and 4) shifting of the cell center leads to a TVD result (Figure 7(b)), whilecell-centered one does not (Figure 6(c)).

Cell sequence j − 2, j − 1, j (see Figure 7(c)). In this case, we have: ∆−xj = (1+a)∆xj

2 and ∆+xj =

[1 + 2b(1 − p)]∆xj

2 . Hence: D−uj =2f∆0uj

(1+a)∆xj, D+uj =

2(1−f)∆0uj

[1+2b(1−p)]∆xjand D0uj =

2∆0uj

(2+a+2(1−p)b)∆xj;

where a = 1, b = 3/4 and p = 1/3. Definition of the ( ϕj−1)TVDDj−1 is the same as the uniform case,Eq. ((8.3)). Comparison of this limiter with the GMINMOD is shown in Figure 7(c). It is obvious, theGMINMOD limiter remains in the TVD region.

Cell sequence j, j + 1, j + 2 (see Figure 7(d)). Here we have: ∆−xj =(1+2ap)∆xj

2 and ∆+xj =(1+b)∆xj

2 . Hence D−uj =2f∆0uj

(1+2ap)∆xj, D+uj =

2(1−f)∆0uj

(1+b)∆xjand D0uj =

2∆0uj

(2+2ap+b)∆xj; where a =

3/2, b = 1 and p = 1/3. The function (ϕj+1)TVDDj+1 is the same as the uniform case. This functionis compared with the GMINMOD limiter in Figure 7(d), where stable and TVD results are attached.

In General, it can be concluded that on irregular grid points with typical grid configuration illustratedin Figure 7(a), numerical solutions will remain TVD and thereby stable.


»xj-1 xj xj+1

xj-32 xj-12 xj+12 xj+32

HaL

xj-2 xj-1 xj xj+1 xj+2

xj-32 xj-12 xj+12 xj+32

D-xj-1=8 dx D-xj=4 dx D-xj+1=2 dxD+xj+1=dx

Dxj-1=6dx Dxj=3dx Dxj+1=1.5dx

H1-pLDxj p DxjH1-pLDxj-1 p Dxj-1

H1-pLDxj+1p Dxj+1

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

ΦiH

u xL i=HD

0uiD

x iL´

HbLa=2b=0.5p=13

u ¢i GMINMOD, Θ=1 GMINMOD, Θ=2; D0ui=D0ui D0xi

Figure 8. Effects of weaker restriction on wavelet-based grid variation. For this figure,we assume xj+1−xj = 0.5(xj−xj−1); a) cell configurations; b) the GMINMOD limiterperformance.

8.8. Constraint on cell center adaptation in the wavelet-based algorithm. As mentioned be-fore, it is always assumed that there exist at least three neighbor cell centers of equal distance fromeach other in non-uniform adapted cell centers. Let assume an adapted cell with weaker cell-sequencecondition: xj+1 − xj = 0.5(xj − xj−1) or xj+1 − xj = 2(xj − xj−1). As an example let consider:xj−1 − xj−2 = 8dx, xj − xj−1 = 4dx, xj+1 − xj = 2dx and xj+2 − xj+1 = dx (Figure 8(a)). For thiscase, if cell edges are assumed to be in the middle of cell centers, xj+1/2 = 0.5(xj + xj+1), then celllengths are: ∆xj−1 = 6dx,∆xj = 3dx and ∆xj+1 = 1.5dx. Hence: ∆−xj = [(1− p) + ap]∆xj , ∆+xj =

[p+ (1− p)b], D−uj =f∆0uj

[(1−p)+ap]∆xj, D+uj =

(1−f)∆0uj

[p+(1−p)b]∆xj, and D0uj =

∆0uj

[1+ap+b(1−p)]∆xj; where a = 2

and b = 0.5. For all cells, shifting coefficient is equal to p = 1/3 (see Figure 8(a)).The GMINMOD (Eq. (8.4)) and u′j (from non-uniform cases: Eq. (8.11)) are compared in Figure

8(b). It is clear that the limiter is slightly outside the TVD domain. For this reason, long term stabilityof numerical solutions could vanish. For cell-centered cases, where xj = (xj+1/2+xj−1/2)/2, the resultsare not also TVD, see Figure 6(b).

To have stable results, cell centers are located in a way that there exists always at least threeneighbor grids (cell centers) of equal distance from each other. For guarantee this condition, in the gridmodification stage (the post-processing stage), it is assumed, at least, to have: Ns = Nc = 1.

9. Numerical examples

The following examples are to study the effectiveness of the proposed method concerning nonlinear 1-D and 2-D first order hyperbolic systems. The main assumptions are: 1- applying the D-D interpolatingwavelet of order 3; 2- using the generalized MINMOD flux/slope limiter in all problems; 4- repeatingre-adaptation processes every time step; 5- using the semi-discrete form of central and central-upwindschemes; 6- integrating in time by the TVD Rung-Kutta second-order solver.Burgers’ equation. The Burgers equation is defined as follows:

ut +1

2

(u2)x= 0,

where u is the conserved quantity and its flux is F (u) = u2/2. The system is nonlinear, so thatdiscontinuous fronts will develop during front propagations. Here it is assumed that the initial andboundary conditions are:

BCs : u(x = 0, t) = u(1, t) = 0, ICs : u(x, t = 0) = sin(2πx) +1

2sin(πx).


0.0 0.2 0.4 0.6 0.8 1.0

-0.5

0.0

0.5

1.0

x

u

HaL t=0.158

t=0.5

t=1

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HbLt=0.158

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HcLt=0.5

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HdLt=1

Figure 9. The Burgers’ solution and corresponding adapted grid points (in figure (a),exact solutions are presented by the solid line and numerical solutions are illustratedby points and hollow shapes).

For above conditions, a discontinuity starts to appear around t ≈ 0.158. This discontinuous frontwill propagate to the right side after this time. Assumptions for the numerical simulations are: ϵ = 10−3

(threshold) and θ = 2 (the flux limiter parameter), Nc = Ns = 2 (for post-processing stage), Jmax = 11,and Jmin = 5. The numerical results are illustrated in Figure 9 at times 0.158, 0.5, and 1. This figurecontains numerical results, exact solutions and corresponding adapted grids in different resolutions.The results confirm that adapted points are properly concentrated around propagating fronts.

The KT central scheme and NVSF-based method [12] are compared on wavelet-based adapted grids(the NVSF-based method is basically developed for non-uniform grid points). The results are presentedin Fig. 10 at time 0.158. For the NVSF formulation, two types of flux limiters are considered: theSMART and MINMOD limiters [12]. For all simulations, we have ϵ = 10−3.

The results offer that the proposed method is comparable with methods originally provided fornon-uniform grid points.Euler system of equations. For this system, the governing equation is:

∂

∂t

ρρuE

+∂

∂x

ρuρu2 + Pu(E + P )

=

000

,

where ρ, u and E are gas density, velocity and total energy, respectively. The pressure P can be obtained

by: P = (γ − 1)(E − ρu2

2

), where γ is the ratio of specific heats, and here it is assumed to be γ = 1.4.

In the following three different problems with different initial and boundary conditions will be studied.These diverse conditions lead to different bench-mark problems; they are: 1) the Sod problem [49]; 2)the Lax problem [33]; 3) Interaction of an entropy sine wave with a Mach 3 right-moving front [48].Sod Problem. Corresponding initial conditions are: ρ

uP

t=0

=

0, 0, 1T , x ≤ 0.5,

0.125, 0, 0.1T , x > 0.5,


0.40 0.45 0.50 0.55 0.600.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

x

u

Exact sol.

NVSF-SMART

NVSF-MINMOD

KT-GMINMODΘ=2

HaLt=0.158

0.55 0.60 0.65 0.70 0.75

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

x

u

HbLt=0.158

Figure 10. Comparison of the proposed adaptive central method and NVSF-basedupwind scheme.

An unbounded 1-D domain is assumed: a Riemann problem. Assumptions for numerical simulationsare: Jmax = 11, Nd = 6, ϵ = 10−3, θ = 2, Nc = 2, Ns = 1 and dt = 10−5. The numerical solutionsand corresponding adapted grid points are illustrated in Figure 11 at time 0.2 with methods KT, M1andM2. For the KT method, results are presented in Figures 11(a)-(c); these figures contains numericalresults, corresponding entropy productions (Sn

j ), adapted grids and local truncation errors (Enj ). These

results for the M1 and M2 schemes are provided respectively in Figures 11(d-f) and Figures 11(g-i).Figure 11 provides that: 1) methods M1 & M2 have less numerical dissipation in comparison with

the KT scheme; 2) the M2 scheme leads to the smallest dissipation; 3) using a less dissipative method,more grid points concentrate automatically in different resolutions. This is confirmed by comparing Ng

values of these three methods during simulations, see Figures 11(b,e,h) and Figure 12.The numerical entropy production cannot detect the contact discontinuity in this example (Figures

11(a),(d) & (g)), even by less dissipative methods with fine enough resolutions. Entropies Snj have

small values in the rarefaction zone (for 0.25 < x < 0.45), but can properly detect shock waves. Thelocal truncation error can capture both the shock wave and contact discontinuity. The local errors En

j

have considerable values in rarefaction zones. This zone is not detected by the wavelet theory; as aresult, grid points do not adapted there. Considering the wavelet-based adapted points, results of Sn

j ,and En

j , different criteria lead to different adapted grids. In this example, it seems that wavelet-basedadaptation method leads to more realistic adapted grids.

Effects of the post-processing stage are investigated by some numerical simulations in the following.At first, effect of considering the post-processing stage is studied. Two simulations with and without thepost-processing stage are done and results are presented in Figure 13. Figures 13(a-b) and Figures 13(c-d) include solutions with and without the post-processing step, respectively. The results indicate thatpost-processing adapted grids have significant effects on solution stability. The numerical instabilitygrows rapidly in absence of the post-processing step.

The post-processing stage contains both grid modification in the same resolution and successivecoarser resolution, see 3. To study effects of them, two types of modifications are considered: full andpartial grid modification (by a post-processing). The modifications are: 1) partial post processing: fora grid point having resolution j, new points are only added at the corresponding resolution level; herewe assume: Ns = 2 & Nc = 0; 2) full post processing: both resolution level j and j + 1 are controlled;we choose: Ns = 2 & Nc = 1. The former, the semi-modification, is frequently used in wavelet-basedadaptation procedures. Numerical results are presented in Figure 14. Figure 14(a) and Figure 14(b)correspond to methods using the full and partial post-processing stages, respectively. This figure showsthat long-term stability can be obtained in case of having full-modification, 14(a).


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x

Ρ&H

S¤2

00L

HaLΕ=10-3

KT

å

Exact Sol.EntropyNumerical sol.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x

Ρ&H

S¤2

00L

HdLΕ=10-3

M1

ë


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x

Ρ&H

S¤2

00L

HgLΕ=10-3

M2

ó


0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HbL

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HeL

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HhL

0.0 0.2 0.4 0.6 0.8 1.0

-2

-1

0

1

2

x

Ejn H

10-

8 L

HcL

0.0 0.2 0.4 0.6 0.8 1.0

-2

-1

0

1

2

x

Ejn H

10-

8 L

HfL

0.0 0.2 0.4 0.6 0.8 1.0

-2

-1

0

1

2

x

Ejn H

10-

8 L

HiL

Figure 11. Numerical results, corresponding truncation errors, entropy productions,and adapted points in different resolutions for the Sod problem with the KT,M1 &M2schemes at t = 0.2; a-c) the KT scheme; d-f) the M1 method; g-i) the M2 scheme.

0.00 0.05 0.10 0.15 0.20 0.2550

100

150

200

250

t

Ng

Ε=10-3

Jmax=11

Nd=6

KT schemeM1M2

Figure 12. Number of adapted grid points Ng during simulations. In the finest reso-lution, number of grid points is 211 + 1.

Lax Problem. The initial conditions are: ρuP

t=0

=

0.445, 0.69887, 3.5277 , x ≤ 0.5,

0.5, 0, 0.571 , x > 0.5,


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x

Ρ

HaLt=0.0012Θ=2

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HbL

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x

Ρ

HcLt=0.0012Θ=2

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HdL

Figure 13. Post-processing effects on stability of solutions; a-b) with the post-processing stage; c-d) without the post-processing step. In figures (a) & (c), solidlines and hollow shapes are exact and numerical solutions, respectively.

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x

Ρ

HaLt=0.25Θ=2

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

x

Ρ

HbLt=0.25Θ=2

Figure 14. Full and semi post processing effects; a) full post processing, Ns = Nc = 2;b) semi-post processing, Ns = 2,&Nc = 0. Solid lines and hollow shapes are exact andnumerical solutions, respectively.

and the problem is a Riemann problem. For simulations, it is assumed: ϵ = 10−3, θ = 2, and dt = 10−5.For the three methods KT, M1 and M2, solutions ρ, corresponding entropies Sn

j , adapted grids,and truncation errors En

j are presented in Figure 15 at time 0.16. Same as the Sod problem, theresults offer that: 1) less dissipative methods mobilize more adapted grid points of fine resolutions; 2)numerical entropy production of these methods cannot detect the contact discontinuity; 3) the localtruncation errors En

j can detect the contact discontinuity zones; 4) Enj can also detect rarefaction zones;

5) the wavelet transform can properly capture all phenomena: shock waves, rarefaction and contactdiscontinuity zones.Interaction of an entropy sine wave with a Mach 3 right-moving front. This challengingproblem was developed to reveal high order scheme capabilities by Shu and Osher [48]. Here, it isassumed the ratio of specific heats is γ = 1.4. The Riemann initial condition is [48, 35]:


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

x

Ρ&H

S¤5

0L

HaLΕ=10-3

KT

å


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

x

Ρ&H

S¤5

0L

HdLΕ=10-3

M1

ë


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

x

Ρ&H

S¤5

0L

HgLΕ=10-3

M2

ó


0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HbL

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HeL

0.0 0.2 0.4 0.6 0.8 1.03

4

5

6

7

8

9

10

x

j

HhL

0.0 0.2 0.4 0.6 0.8 1.0

-2

-1

0

1

2

x

Ejn H

10-

8 L

HcL

0.0 0.2 0.4 0.6 0.8 1.0

-2

-1

0

1

2

x

Ejn H

10-

8 L

HfL

0.0 0.2 0.4 0.6 0.8 1.0

-2

-1

0

1

2

x

Ejn H

10-

8 L

HiL

Figure 15. Numerical results, corresponding entropy production, truncation errors,and adapted points of different resolutions for the Lax problem at t = 0.1; a-c) the KTscheme; d-f) the M1 method; g-i) the M2 scheme.

ρuP

t=0

=

3.857143, 2.629369, 10.33333 , x < −4,

1 + 0.2sin(5x), 0, 1 , x ≥ −4.

The considered computational domain is: Ω ∈ (−5, 5) × (0, T ); Assumed parameters are: ϵ = ϵ0 =5× 10−3, Jmax = 11, Jmin = 5 (or Nd = 6), Nc = 1, Ns = 2, θ = 2, dt = 0.00025.

The numerical entropy production, numerical and exact solutions are illustrated in Figure 16 at t =1.8. There, the solid lines and hollow shapes are the reference [35] and numerical solutions, respectively.Regarding numerical entropy productions, it is clear that both the M1 and M2 methods lead to lessnumerical dissipations in comparison to the KT scheme. The M2 scheme leads to the least dissipativeresults, since magnitude of entropy S is larger than both the KT and M1 methods. Distribution ofadapted points at different resolution levels for these three schemes are shown in Figure 17 at t = 1.8.The methodsM1 andM2 lead to more adapted points of high resolutions. In Figure 18 local truncationerrors for these three methods are presented, which confirm numerical convergence.2-D Euler equation of gas dynamics for ideal gases. The governing equation for the 2-D systemis:

ut + Fx +Gy = 0,

where state values are u = ρ, ρu, ρv, ET ; the flux vectors in the x and y directions are F =ρu, ρu2 + P, ρuv, u(E + P )

Tand G =

ρv, ρuv, ρv2 + P, v(E + P )

T, respectively.


-4 -2 0 2 40

1

2

3

4

5

xΡ

&H

S¤1

0L

HaLΕ=0.510-3

Jmax=11KT

:

-4 -2 0 2 40

1

2

3

4

5

x

Ρ&H

S¤1

0L

HbLΕ=0.510-3

Jmax=11M1

,

-4 -2 0 2 40

1

2

3

4

5

x

Ρ&H

S¤1

0L

HcLΕ=0.510-3

Jmax=11M2

>

Figure 16. Numerical results, and entropy productions for the right propagating frontwith the KT, M1 & M2 schemes at t = 1.8. In these figures, solid black lines are thereference solutions, hollow shapes are numerical ones, and the gray solid lines arenumerical entropy productions.

-4 -2 0 2 43

4

5

6

7

8

9

10

x

j

HaLΕ=0.510-3

Jmax=11KT

:

-4 -2 0 2 43

4

5

6

7

8

9

10

x

j

HbLΕ=0.510-3

Jmax=11M1

,

-4 -2 0 2 43

4

5

6

7

8

9

10

x

j

HcLΕ=0.510-3

Jmax=11M2

>

Figure 17. Distribution of adapted grid points in different levels of resolution for theKT, M1 & M2 schemes at t = 1.8.

In this example, radially symmetric initial conditions are assumed with respect to the origin. Thereexists initial higher density and higher pressure inside a circle with radios r = 0.4; corresponding valuesare: ρin = 1, ρout = 0.1 & Pin = 1, Pout = 0.1. Other initial values are: uin = uout = vin = vout = 0[31, 54]. The computation domain belongs to Ω ∈ [−1.5, 1.5]× [−1.5, 1.5]. To control the symmetric ofsolutions and corresponding adapted grids in simulations, total of the computing domain Ω is considered.


-4 -2 0 2 4

-0.4

-0.2

0.0

0.2

0.4

x

Ejn H

10-

5 L

HaLΕ=0.510-3

Jmax=11KT

:

-4 -2 0 2 4

-0.4

-0.2

0.0

0.2

0.4

x

Ejn H

10-

5 L

HbLΕ=0.510-3

Jmax=11M1

,

-4 -2 0 2 4

-0.4

-0.2

0.0

0.2

0.4

xE

jn H

10-

5 L

HcLΕ=0.510-3

Jmax=11M2

>

Figure 18. Local truncation errors for the right propagating front with the KT, M1& M2 schemes at t = 1.8.

The numerical results are presented in Figure 19 at t = 0.4. Figures 19(a) & (b) are from the KTscheme and Figures 19(c) & (d) belong to theM1 method. The results offer that: 1) all of the solutionsand corresponding adapted grids are symmetric; 2) due to numerical dissipation, the KT solver leads toslightly different result from the M1 one. To clarify the numerical dissipation effects, cut of solutionsare compared along y = 0, Figure 20. This figure confirms that the KT scheme ends to more dissipativeresults.

The local truncation errors Enj for the two schemes (KT andM1) and corresponding adapted grids are

presented in Figure 21 at t = 0.2. It is clear that the errors are properly concentrated in high-gradientzones detected properly by the wavelet transform.A non-convex example: the polymer system. The governing equation of the polymer system is:

∂

∂t

(sb

)+

∂

∂x

(f(s, c)cf(s, c)

)=

(00

),(9.1)

where s denotes water saturation; parameter c is the polymer concentration; function f := f(s, c)presents the fractional flow function of water; parameter b is function of s and c where b := b(s, c).

Functions b and f are assumed to be: b(s, c) = sc + a(c) and f(s, c) = s2

s2+(0.5+c)(1−s)2 , where a(c)

denotes the adsorption function and in this example, it is: a(c) = c5(1+c) .

The eigenvalues of the polymer system ((9.1)) are: λ1 = fs(s, c) and λ2 = f(s, c)(s+ a′(c))−1.For numerical simulations, we assume: Jmax = 11, Nd = 6, ϵ = 10−4, dt = 0.00025, and Ns = Nc = 1

(for grid modification).In numerical simulations two types of θ are assumed: constant and adaptive. In adaptive case, in

this work, it is assumed θ depends linearly on spatial positions of cell centers xj , as:

θ(xj) = 1 +(∆xj +∆xj+1) /2 −∆xmin

∆xmax −∆xmin, θj := θ(xj),

where ∆xj := xj − xj−1, ∆xmin := min ∆xj, and ∆xmax := max ∆xj. So, around high-gradientsolutions θ → 1 and in smooth regions θ → 2. To study resolution effect, this example is also re-simulated for a fine resolution with resolution number: Jmax = 13. In this case, number of decompositionlevels is Nd = 8 and the time step is chosen in such a way that the CFL number does not change.

Numerical results and corresponding local truncation errors for the parameter s are illustrated inFigure 22. The results provide that: 1) for case θ = 2, the numerical solution does not converge to real


Figure 19. The 2-D Euler gas dynamic adaptive solutions with corresponding adaptedgrids at t = 0.4.

-1.5 -1.0 -0.5 0.0 0.5 1.0 1.50.10

0.15

0.20

0.25

0.30

x

Ρ

M1KT

Figure 20. Comparison of numerical results obtained by the KT and M1 schemesalong y = 0 at t = 0.4.

one, even though it is a converged weak solution (Figure 22(b)); 2) by using an adaptive scheme withadaptive θ, the result is nearly in accordance with the real solution, Figure 22(c); by increasing Jmax

values (or using finer resolutions), the numerical solution approaches to the reference one, Figure 22(e);4) all of the results are the converged weak solutions due to errors En

j ; 5) using more higher resolutionlevel Jmax, smaller local truncation errors are.


Figure 21. Local truncation errors for the KT and M1 schemes at t = 0.214.

1.0 1.2 1.4 1.6 1.8 2.00.2

0.3

0.4

0.5

0.6

0.7

0.8

x

s

HaLΕ=10-4

Jmax=11

Nd=6

é

Exact

KT schemeΘ=2

1.0 1.2 1.4 1.6 1.8 2.00.2

0.3

0.4

0.5

0.6

0.7

0.8

x

s

HcLΕ=10-4

Jmax=11

Nd=6

é

Exact

KT schemeAdaptive Θ

1.0 1.2 1.4 1.6 1.8 2.00.2

0.3

0.4

0.5

0.6

0.7

0.8

x

s

HeLΕ=10-4

Jmax=13

Nd=8

é

Exact

KT schemeAdaptive Θ

1.0 1.2 1.4 1.6 1.8 2.0

-0.10

-0.05

0.00

0.05

0.10

x

Ejn H

10-

6 L

HbLΕ=10-4

Jmax=11

Nd=6

KT scheme

Θ=2

1.0 1.2 1.4 1.6 1.8 2.0

-0.10

-0.05

0.00

0.05

0.10

x

Ejn H

10-

6 L

HdLΕ=10-4

Jmax=11

Nd=6

KT scheme

Adaptive Θ

1.0 1.2 1.4 1.6 1.8 2.0

-0.10

-0.05

0.00

0.05

0.10

x

Ejn H

10-

6 L

HfLΕ=10-4

Jmax=13

Nd=8

KT scheme

Adaptive Θ

Figure 22. Numerical solutions, corresponding adapted grids and local truncationerrors for the polymer system; a, b) with θ = 2; c,d) with adaptive θ; e,f) adaptive θ.

2-D scalar conservation laws with non-convex fluxes. The assumed conservation law is:

ut + sin(u)x + cos(u)y = 0,

where u := u(x, y, t) and the initial condition is:

u(x, y, t = 0) =

14π4 , x2 + y2 < 1,

π4 , x2 + y2 ≥ 1.

For numerical simulations, we have: Jmax = 8, Jmin = 5 (or Nd = 3), ϵ = 10−4, dt = 0.5 × 10−3,and Ns = Nc = 1 (for modification of adapted grid ). For modeling, two different choices of θ areassumed: 1) the constant one with value θ = 2; 2) the adaptive implementation of θ. The latter is alsobased on the 1-D linear interpolation of θ on adapted grid points, as:


Figure 23. Adaptive solutions of 2-D non-convex conservation law system at t = 1;a, b) obtained with θ-adaptive KT scheme; c, d) based on the KT scheme with θ = 2.

θ(Zj) = 1 +(∆Zave)j −∆Zmin

∆Zmax −∆Zmin,

where (∆Zave)j := (∆Zj +∆Zj+1) /2, ∆Zj := Zj − Zj−1, Zj ∈ xj , yj, ∆Zmax = 1/2Jmin ,

and ∆Zmin = 1/2Jmax . For each direction, θ(Zj) is calculated independently.The numerical results and corresponding adapted grid points are shown in Figure 23 at t = 1.

Figures (a, b) correspond to the θ-adaptive results and figures (c, d) are from the constant θ. Againthe θ-adaptive solver converges to proper and physical results [31].

10. Conclusion

In this study, a wavelet-based adaptation procedure is properly integrated with central/central-upwind high resolution schemes for simulation of first order hyperbolic PDEs. It is shown that centralhigh-resolution schemes become unstable on non-uniform cells even those have gradual grid densityvariations. This is because, the NVSF criterion is not satisfied by central schemes. Since their slope/fluxlimiters do not remain TVD on irregular cell-centered cells. Two key ideas are followed to remedy theinstability problem: 1) replacing local irregular grids with abrupt changing with grids having gradualvariations (replacing an ill-posed problem with a nearly well-posed one); 2) studying performance oflimiters on irregular cells. The grid modification stage is done in framework of multiresolution analysis.The TVD conditions are reviewed and provided for non-uniform cells. It is shown that on cell-centeredcells, limiter definitions should be modified. Another approach is using of non-cell-centered cells. Inthis case common limiters can be used without modification. The TVD conditions are derived for bothcell-centered and non-cell-centered cells. Based on these conditions, proper configuration of adapted


cells and corresponding cell-centers are derived. It is shown that cell-center shifting is necessary onlyin some cells acting as transmitting cells. They connect surrounding uniform cells with different celllengths.

The local truncation errors for 1-D and 2-D problems are provided on non-uniform cells for con-vergence studying. Also concept of the numerical entropy production is investigated for uniquenessinsurance. These two concepts have also been used as criteria for grid adaptation. In this regard,performance of these concepts are compared with the wavelet-based algorithm. It is numerically shownthat: 1) the numerical entropy production can not detect some phenomena such as contact disconti-nuities; 2) this concept also have enough values in rarefaction domains. T his can lead to unnecessaryconcentration of grid points in such regions (this can also be seen in [17, 45]); 3) it seems performanceof the local truncation errors as detector is better than the numerical entropy production; 4) waveletscan properly detect all of the shock waves, rarefaction regions, and contact discontinuities.

Non-linear hyperbolic systems with non-convex fluxes are studied. It is shown by local truncation er-rors that both physical and non-physical solutions converge to weak solutions. Both method (flux/slopelimiters) and grid adaptations are used for capturing physical solutions.

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Institute of Structural Mechanics (ISM), Bauhaus-University Weimar, 1599423 Weimar, GermanyCurrent address: Institute of Structural Mechanics (ISM), Bauhaus-University Weimar, 1599423 Weimar, Germany

E-mail address: [email protected]; [email protected]

Department of Mathematics, Center for Scientic Computation and Mathematical Modeling, (CSCAMM),Institute for Physical Science and Technology (IPST), University of Maryland, College Park, MD 20742-

4015, USAE-mail address: [email protected] (Coresponding Author)

Institute of Structural Mechanics (ISM), Bauhaus-University Weimar, 1599423 Weimar, Germany andSchool of Civil, Environmental and Architectural Engineering, Korea University, Seoul, South Korea

E-mail address: [email protected]

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