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SIAM J. SCI. COMPUT. c© 2016 Society for Industrial and Applied MathematicsVol. 38, No. 2, pp. A691–A711

HYBRID COMPACT-WENO FINITE DIFFERENCE SCHEME WITHCONJUGATE FOURIER SHOCK DETECTION ALGORITHM FOR

HYPERBOLIC CONSERVATION LAWS∗

WAI-SUN DON† , ZHEN GAO‡ , PENG LI§ , AND XIAO WEN‡

Abstract. For discontinuous solutions of hyperbolic conservation laws, a hybrid scheme, basedon the high order nonlinear characteristicwise weighted essentially nonoscillatory (WENO) conser-vative finite difference scheme and the high resolution spectral-like linear compact finite difference(Compact) scheme, is developed for capturing shocks and strong gradients accurately and resolvingsmooth scale structures efficiently. The key issue in any hybrid scheme is the design of an accurate,robust, and efficient high order shock detection algorithm that is capable of determining the smooth-ness of the solution at any given grid point. The conjugate Fourier (cF) partial sum and its derivativeare investigated for its applicability as a shock detector due to its unique property, namely, the cFpartial sum converges to the location and strength of an isolated jump. For a nonperiodic problem,the data are first evenly extended before the derivative of the cF partial sum and its mean are com-puted. The mean allows one to partition the domain into subdomains containing strong gradientsor smooth solutions. The locations of shocks are then accurately identified and flagged for specialtreatment using the WENO scheme. The matrix-multiply, even-odd decomposition, and cosine/sinefast transform algorithms of the cF analysis are derived, and their advantages and disadvantages intheir implementations, usage, and technical issues are discussed in detail. The cF shock detector andits iterative version for detecting jumps of large difference in scales are presented. The Hybrid-cFscheme is applied to one-dimensional shock-density wave interaction problem, two-dimensional (2D)Riemann initial value problems, and 2D Mach 10 double Mach reflection problems. The preliminaryresults are in good agreement with those obtained in the literature. They demonstrate the spatialand temporal adaptivity of the Hybrid-cF scheme for problems containing strong shocks, multipledeveloping shocklets, and high frequency waves. A speedup of the CPU times by a factor up to 2–3is obtained showing the potential efficiency of the Hybrid-cF scheme over the pure WENO-Z scheme.

Key words. conjugate Fourier, WENO, compact, hybrid, hyperbolic, shock detection, Riemann

AMS subject classifications. 65P30, 77Axx

DOI. 10.1137/15M1021520

1. Introduction. The nonlinear system of hyperbolic conservation laws (PDEs)can be written compactly as

(1.1)∂Q

∂t+∇ ·F(Q) = 0,

where Q and F(Q) are the conservative variables and flux, respectively, together withappropriate initial conditions and boundary conditions in a Cartesian domain. It iswell known that the solution of (1.1) can develop finite time singularities, such asshocks, contact waves, and rarefaction waves, at some later time even if the solution

∗Submitted to the journal’s Methods and Algorithms for Scientific Computing section May 18,2015; accepted for publication (in revised form) December 1, 2015; published electronically March1, 2016. This work was partially supported by NNSFC (11201441,11325209,11221202), CPSF(2012M521374, 2013T60684), Fundamental Research Funds for the Central Universities (201362033),and the startup fund (201412003) at Ocean University of China.

http://www.siam.org/journals/sisc/38-2/M102152.html†Corresponding author. School of Mathematical Sciences, Ocean University of China, Qingdao,

266100 China ([email protected]).‡School of Mathematical Sciences, Ocean University of China, Qingdao, 266100 China

([email protected], xiaowen [email protected]).§State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology,

Beijing, China ([email protected]).

A691

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mailto:[email protected]





A692 WAI-SUN DON, ZHEN GAO, PENG LI, AND XIAO WEN

is a smooth function initially. The solution of such nonlinear systems could also createboth fine complex smooth and large strong gradient flow structures dynamically inspace and time.

Characteristicwise weighted essentially nonoscillatory (WENO) conservative finitedifference nonlinear schemes on an equidistant stencil, as a class of high order/highresolution nonlinear schemes for solutions of hyperbolic conservation laws in the pres-ence of shocks and small scale structures, was initially developed by [19] (for detailsand history of the WENO scheme, see [2, 3, 33] and references therein). However,there are certain disadvantages of using WENO schemes for some classes of hyperbolicconservation laws. The WENO scheme is fairly complex to implement and compu-tationally expensive. In order to guarantee essentially nonoscillatory capturing ofshocks and resolving high gradient structures, the scheme requires the setup of theRoe-averaged eigensystem, the positive and negative splitting of the fluxes, the for-ward and backward projections of split fluxes between the characteristic and physicalspaces, the computation of the smoothness indicators, and nonlinear weights for eachprojected flux. (Parallel implementation of the WENO scheme will certainly help inreducing the overall CPU time.) These steps make the WENO scheme at least four tofive times more expensive than other nonlinear shock capturing schemes (for example,TVD, PPM). In addition, the WENO scheme is, in general, too dissipative for certainclasses of problems (for example, compressible turbulence) [18, 20] and has limited itswider adoption in its pure form.

A natural remedy to alleviate some of these difficulties is to use a linear schemein place of the nonlinear WENO scheme in smooth regions of the solution whereverpossible. Some often used linear schemes are the central scheme [5, 11] that has astrong dispersive error, the bandwidth optimized central scheme that increases theresolution at a cost of a reduced order of accuracy, and the compact scheme (Compact)that requires solving a system of banded matrix equations [1, 31]. Therefore, a spatialand temporal switching between a WENO scheme for nonsmooth stencils and a linearscheme for smooth stencils (hybridization) must be designed by employing a measureof smoothness of the solution at each grid point and time. In the last two decades,many shock detection methods have been developed. Li, Lu, and Qiu [26] and Liand Qiu [27] showed the comparison among several different low order smoothnessindicators used in a hybrid upwind-WENO scheme for solving hyperbolic conservationlaws and shallow water equations. Costa and Don [5, 6] described the arbitrary ordermultiresolution (MR) analysis by Harten [17] for identifying the non-smooth andsmooth stencils. In general, high order MR analysis is preferable to a low order shockdetector in the high order hybrid scheme for its efficiency (with its O(N) floatingpoint operations count) and much improved resolving power in differentiating highfrequency waves and discontinuities.

By treating a discontinuity as an edge of an object in an image, tools for edgedetection in image processing can be directly applied as well. In [12] and their laterworks, Gelb and Tadmor described the theoretical justification and an algorithm,based on the conjugate Fourier (cF) partial sum with concentration factors, for edgedetection of a piecewise smooth function given its Fourier coefficients.

In a limit, the cF partial sum converges to the associated jump function1 at a rateof O(1/ logN), where N is the number of Fourier modes. A few numerical techniques,such as modifying the concentration factor and nonlinear enhancement procedures,

1The jump function (defined formally in Lemma 2.1) is identically zero everywhere, except atdiscontinuity locations, where it is equal to the jump height.

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HYBRID SCHEME WITH CONJUGATE FOURIER JUMP DETECTOR A693

were developed to accelerate the convergence rate, to remove oscillations, and to re-cover edges in a piecewise smooth function [12, 14]. The method has also been appliedto solving one-dimensional Euler equations using a Legendre collocation method with(adaptive) spectral vanishing viscosity [13, 36]. Jung [21] deployed an iterative algo-rithm based on the modification of the scaling coefficient ε in the multiquadric radialbasis functions to approximate a piecewise discontinuous function, thus avoiding theGibbs phenomenon. To our knowledge, the methods described above are mainly usedin the detection of edges and none other than the high order MR algorithm has beenapplied in solving hyperbolic conservation laws with hybrid schemes. In this study,we will investigate the applicability of the cF method as the shock detector in thehigh order hybrid Compact-WENO scheme for solving hyperbolic PDEs and discussits accuracy, efficiency, and other implementation issues. The algorithmic implemen-tation of the cF method (matrix-multiply (MXM) algorithm, even-odd decomposition(EOD) algorithm, and cosine/sine fast transform (CFT) algorithm), and an effectivestrategy that works in partitioning the domain and detecting edges are given in detail.Numerical examples of the one-dimensional extended shock-density wave interactionproblem, two-dimensional Riemann initial value problems, and Mach 10 double Machreflection (DMR) problems are presented. This study shows the potential of the cFmethod and can spur further development of these techniques in the hybrid schemefor high speed shocked flows.

This paper is organized as follows. In section 2, we briefly review the cF methodand describe the cF edge detection algorithm in detail. Examples are given for edgedetection of the Shepp–Logan phantom image and the efficiency of various cF algo-rithms are presented. In section 3, the one- and two-dimensional shocked problemsare simulated to investigate the performance (accuracy and speed) of the Hybrid-cFschemes. Concluding remarks are given in section 4.

2. Hybrid scheme. In this study, an improved fifth order WENO-Z conserva-tive finite difference nonlinear scheme and a high order spectral-like (Compact) schemeare employed for solutions of multidimensional system of hyperbolic conservation lawsin the Cartesian domain discretized with a uniformly spaced grid.

For the fifth order WENO-Z5 finite difference nonlinear scheme used here, wetake the sensitivity parameter ε = Δx4 with power parameter p = 2 [7]. For thesixth order compact finite difference linear scheme, see [25] for details. We shouldnoted that the derivative of fluxes at the boundary for the boundary closure of theCompact scheme is computed via the WENO-Z5 scheme. The temporal scheme is thethird order TVD Runge–Kutta scheme [33]. Readers are referred to [2, 3, 25, 33] andreferences therein for further details regarding their algorithmic implementations.

The key issue in any spatially adaptive scheme is how to quantify the smoothnessof a numerical solution at each uniformly spaced grid point xi. Since both the Com-pact scheme and the WENO scheme are high order/resolution schemes, the measureof the smoothness of the solution must also be of high order in order to differentiate ahigh frequency wave from a high gradient/shock. The high order information obtainedallows one to apply an appropriate numerical spatial scheme (such as the Compactscheme for high-frequency waves and the WENO scheme for shocks) adaptively at agiven spatial location x and dynamically at a given time t.

Two high order shock detection algorithms on a uniform grid are considered,namely,

• the Lagrangian polynomial based MR analysis, and• the trigonometric polynomial based cF analysis.

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The Lagrangian polynomial based MR analysis [17] has been extensively studiedand successfully applied to the hybrid schemes [5, 6]. The high order trigonometricpolynomial based cF analysis together with a high order concentration factor has beenused for detection of discontinuities in functions and edges in images [12, 14, 15, 35]but seldom used in the context of numerical PDEs. In [15], theoretical developmentof the shock detection using continuous and discrete cF series was reviewed in detail.Below, we will briefly introduce the idea of cF analysis and then develop a shockdetector to identify shocks and discontinuities in the solution of hyperbolic PDEs.Readers are referred to [9, 10, 29] for the detailed theory of the cF series.

2.1. cF analysis. A truncated Fourier series expansion of a 2π-periodic functionf(x) ∈ L2[0, 2π) with 2N + 1 terms is

fN(x) =1

2a0 +

N∑k=1

(ak cos kx+ bk sinkx),(2.1)

where the Fourier coefficients ak and bk are defined as

ak =1

π

∫ π

−π

f(x) cos kx dx, bk =1

π

∫ π

−π

f(x) sin kx dx,(2.2)

respectively, and its corresponding cF series is

fN(x) =

N∑k=1

(ak sin kx− bk cos kx).(2.3)

They are the Fourier partial sum and cF partial sum, or simply Fourier and cF,respectively, for short when their meanings are clear in the context of the statement.

It is well known that the finite sum of the Fourier series in the Fourier expansionof a piecewise smooth function generates O(1) numerical oscillations in the neighbor-hood of a discontinuity (Gibbs phenomenon). The ≈ 8.9% overshoot and undershootof the jump do not decay even as the number of Fourier modes N increases. The nu-merical oscillations often destabilize numerical schemes for solving a time-dependenthyperbolic PDE and reduce the high order scheme to first order at most. There weremany past studies in post processing the oscillatory data to recover spectral accuracyof the solution when solved with spectral methods (see [16, 24] and website2 for re-search that has been done in the past decades) with some degree of success. For thehybrid scheme we are developing, our goal is to treat the discontinuities as edges andto design a simple yet effective procedure to detect and to identify discontinuities,including discontinuities in its first derivative, using the cF and/or its derivative.

The key aspect of the procedure lies in the relationship between the cF and thelocation and jump of a discontinuity. We recall the support of the cF fN(x) (2.3)approaches the singular support of a piecewise smooth function f(x) as N → ∞ asshown in the following lemma [12].

Lemma 2.1. Let f(x) be a 2π-periodic piecewise smooth function, except at asingle jump discontinuity at x = ξ with an associated jump

[f ](ξ) := f(ξ+)− f(ξ−).(2.4)

2http://www.cscamm.umd.edu/people/faculty/tadmor/Gibbs phenomenon/

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Then

− π

logNfN(x)→ [f ](x)δξ(x) =

{[f ](ξ), x = ξ,0, otherwise,

(2.5)

where δξ is the Dirac distribution located at ξ.

This is known as the concentration property of fN(x). Readers are referred to[12] for details of the proof. It can be seen in Lemma 2.1 that the convergence rateis only O(1/(logN)). The convergence rate can be increased to O(logN/N) [12]and faster in smooth regions away from discontinuities using a high order admissibleconcentration factor. A low pass filter and a minmod algorithm can be added to thecF together with a concentration factor to suppress oscillations near a jump and awayin smooth regions, and that significantly reduces the chance of false identification ofa discontinuity near a jump at an expense of enlarging the support around it.

In this work, we employ the hybrid finite difference scheme in solving the hyper-bolic conservation laws on a uniformly spaced grid. The continuous Fourier coefficientswill be approximated by the discrete Fourier coefficients with the data being sampledon a uniformly spaced grid domain with grid spacing Δx using N + 1 grid points(see below). For simplicity, we shall use the same notations ak and bk as the discreteFourier coefficients in the following discussion.

For problems containing high frequency waves and shocks, a high-pass filter willbe employed to maintain the effectiveness of the cF shock detection algorithm (seebelow). The variable order high-pass exponential filter

σk =

{1− exp

(−α

∣∣∣ k−k0

N−k0

∣∣∣γ), |k| > k0,

0, |k| ≤ k0,(2.6)

where k0 = 0 is the cutoff wave number, α = − ln ε such that σN = 1 − ε (typically,ε = 10−16 is the machine zero), and γ = 16 is the order of the filter. Note that, ifno high-pass filter is used, σk = 1. A high-pass filter removes waves with low- andmid-frequency from the data while keeping all the high frequency waves mostly intact.With a carefully chosen filter order γ, the filter will reduce the shock strength slightly,but it will not affect the location of the shocks. In practice, we find that it is moreconvenient and effective in detecting a location of a jump discontinuity of a functionand its first and higher derivatives by using the derivative of the cF,

fN(x) = (fN)′(x) =

N∑k=1

k(ak cos kx+ bk sinkx).(2.7)

Together with a high-pass filter, one has the final form

fN(x) =

N∑k=1

τk(ak sin kx− bk cos kx), τk = σkSk,(2.8)

fN(x) =N∑

k=1

τk(ak cos kx+ bk sin kx), τk = kσkSk,(2.9)

where Sk = sin(kΔx2 )/(kΔx

2 ), and τk and τk are the concentration factors using the

notation of Gelb and Tadmor [12]. An example of the cF fN(x) and its derivative

fN(x) of a square function f(x) with increasing resolution N = 20, 40, 80 is shown in

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x

f(x)

-1 -0.5 0 0.5 10

0.2

0.4

0.6

0.8

1

N=80N=40N=20

x

cF(x

)

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

N=80N=40N=20

x

cF’(x

)

-1 -0.5 0 0.5 1-30

-20

-10

0

10

20

30

N=80N=40N=20

Fig. 1. (Left) The square function f(xi), (middle) its cF ˜fN (xi), and (right) its derivative of

cF fN (xi) with N = 20, 40, 80.

Figure 1. The locations and jumps of the discontinuities of the square function are wellcaptured and estimated with high accuracy. In this work, we are concerned mainlyabout obtaining accurate locations of high gradients and discontinuities so that theWENO reconstruction procedure can capture shocks in an essentially nonoscillatorymanner. Unless noted otherwise, we will only employ the high-pass filter σk when itis absolutely needed.

When solving PDEs, the solutions are often nonperiodic in space. In order toapply the cF shock detection algorithm as described below to such data, we firstperform an even extension of the solution. In this case, the odd discrete Fouriercoefficients are identically equal to zero, that is, bk = 0, due to the even symmetricproperty of the extended solution in an evenly extended domain. As the result, wewill only need the even discrete Fourier coefficients ak in computing the discrete cFand/or its derivative, which can be computed using either the MXM, EOD, or CFTalgorithms, as described below.

2.1.1. MXM algorithm. The discrete Fourier coefficient ak, with the datafi = f(xi) sampled at the Fourier Gauss–Lobatto collocation points xi = πi/N, i =0, . . . , N , becomes

ak =2

Nck

N∑j=0

1

cjfj cos kxj ,(2.10)

where cj = 1, j = 1, . . . , N − 1, and c0 = cN = 2. By substituting (2.10) into (2.3)and (2.9), one has

fN = S�f, fN = D�f,(2.11)

where �f is a vector of length N + 1 with elements fi, and the elements of the(N + 1)× (N + 1) matrices S and D are, respectively,

(2.12) Si,j =2

Ncj

N∑k=1

τkck

cos(kxj) sin(kxi), Di,j =2

Ncj

N∑k=1

τkck

cos(kxj) cos(kxi).

This matrix-multiply approach, with O(N2) floating point operations, is calledthe MXM algorithm and can be represented graphically as

�fS−→ fN , �f

D−→ fN .(2.13)

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2.1.2. EOD algorithm. It is easy to see that S (D) is a centro-anti-symmetric(centrosymmetric) matrix with Si,j = −SN−i,N−j (Di,j = DN−i,N−j). By exploitingthe centro-anti-symmetric properties of S, we can devise an efficient algorithm knownas the EOD algorithm [8, 34] to reduce the floating point operations count by nearlyhalf, thus speeding up the CPU computation time by up to 35% when compared withthe MXM algorithm.

Assuming that N is even, for i, j = 0, . . .N/2, one has

(2.14) fN(xi) =

N/2−1∑j=0

(Di,j +Di,N−j)ej +

N/2−1∑j=0

(Di,j −Di,N−j)oj +Di,N/2eN/2,

where the even ej and odd oj functions are defined as

ej =fj + fN−j

2, oj =

fj − fN−j

2, j = 0, . . . , N/2,(2.15)

respectively. By defining two smaller even E and odd O matrices of size (N/2+ 1)×(N/2 + 1), with elements

(2.16) Ei,j =

{Di,j +Di,N−j ,Di,N/2,

Oi,j =

{Di,j −Di,N−j if j = 0, . . . , N/2− 1,0 if j = N/2,

(2.14) can be written compactly as

fN(xi) = E�e+O�o, i = 0, . . . , N/2,(2.17)

fN(xi) = E�e−O�o, i = N/2, . . . , N.(2.18)

A similar procedure can be devised for odd N (see [4]) and for computing the cF ofa function.

This even-odd decomposition approach, with O(N +N2/2) floating point opera-tions, is called the EOD algorithm and can be represented graphically as

�f →{

�eE−−→ fe

�oO−−→ fo

}→ fN , �f →

{�e

O−−→ fe

�oE−−→ fo

}→ fN .(2.19)

2.1.3. CFT algorithm. For a large number of grid points N as often needed innumerical PDE, both the MXM and EOD algorithms are not only computationallyexpensive in setting up the matrices and require a large amount of memory for storingthe large matrices but also slow in computing the cF and/or its derivative. A more

efficient algorithm can be developed by recognizing that both cF fN and its derivativefN can be computed via the CFT algorithm [8], which would result in a substantial

saving in CPU time for sufficiently large N . Recalling that, given a data vector �fwith length N + 1, the discrete even Fourier coefficients

ak =2

Nck

N∑j=0

1

cjfj cos kxj ,(2.20)

with cj = 1, j = 1, . . . , N − 1, and c0 = cN = 2, can be computed efficiently bythe cosine fast transform algorithm, provided that the transform length N consistsof a product of small prime numbers raised to some powers (for example, N =2k13k25k37k4 · · · ), and is most efficient if N = 2k1 . Next, the Fourier coefficientsak are modified by τk (τk) to obtain the modified Fourier coefficients, for example,

ak = τkak (ak = τkak). Finally, fN (fN) can be found by the inverse sine (cosine) fasttransforms with coefficients ak (ak).

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N

CP

U T

ime

102 103 10410-6

10-5

10-4

10-3

10-2

10-1

MXMEODCFT

Fig. 2. The CPU timings (in seconds) of the cF analysis with the MXM, EOD, and CFTalgorithms as a function of number of grid points N .

This fast Fourier transform approach, with O(5N logN) floating point operations,is called the CFT algorithm and can be represented graphically as

�fCFT−−−→ ak

×τk−−−→ akInverse SFT−−−−−−−→ fN ,(2.21)

�fCFT−−−→ ak

×τk−−−→ akInverse CFT−−−−−−−→ fN .(2.22)

The CPU timings (in seconds) of the cF analysis with the MXM, EOD, and CFTalgorithms as a function of grid points (resolution) N are shown in Figure 2. Forsmall N < 256, all three algorithms spend almost the same amount of CPU time withthe CFT algorithm being the fastest among them. For large N > 256, the CPU timesconsumed by the MXM and EOD algorithms grow like O(N2) with the EOD algorithmbeing the faster between them. The CPU timing of the CFT algorithm, on the otherhand, grows like O(N logN) and is substantially smaller than the CPU timing of boththe MXM and EOD algorithms. Hence, the CFT algorithm, though complex in itsimplementation, is most efficient and recommended for large scale computing.

2.1.4. Remarks, comments, and cautions. Before we proceed to describe thecF shock detection algorithm, we offer the following remarks regarding the algorithmsand their applicability.

Remark 2.2. The MXM algorithm is most straightforward to implement, whilethe EOD algorithm is only slightly more complicated. The CFT algorithm, requiresthe most programming effort and the use of an optimized fast Fourier transform(FFT) library. For a casual user of the cF analysis and/or a small problem size N ,the MXM algorithm is probably the best choice for the ease of programming andprototyping. Otherwise, the CFT algorithm probably is preferred if efficiency andspeed of computing are issues.

Remark 2.3. The MXM and EOD algorithms have flexibility that allows one tocompute the cF or its derivative at any given grid point, while the CFT algorithmrequires one to compute the cF and/or its derivative at all N + 1 grid points all atonce.

Remark 2.4. Both MXM and EOD algorithms employ the BLAS 3 level sub-routine DGEMM to perform the matrix-matrix multiply operation, while the CFT al-gorithm uses the vectorized version of the FFT package VFFTPack written by P. N.Swarztrauber of NCAR. These subroutines as well as the compact scheme, the WENO

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scheme, the cF analysis, and the MR analysis are all included in the high performancesoftware library HOPEpack developed by W. S. Don and his collaborators.

Furthermore, all the computations presented in this study are performed on adesktop computer with an 8 core i7-2600 processor and 8 GB memory running underthe CentOS 6 operating system.

Remark 2.5. As a programming note, unless N or the high-pass filter changes,the matrices S and D of the MXM algorithm (E and O of the EOD algorithm)can be computed once and stored for later use. Otherwise, these matrices must berecomputed before their use.

Remark 2.6. A major caveat of an even extension of an arbitrary nonperiodicfunction, for example, a linear function, is that the first or higher derivatives of thefunction at the boundary might not be continuous in an evenly extended domain. Thederivative of the cF at the boundary, though small, will not be zero, even if the functionis smooth. This might cause the cF shock detection algorithm to misclassify theboundary point as a nonsmooth stencil (depending on some user defined parameters).To remedy this scenario, for example, in a domain [x00, x01]× [y00, y01], one can firstpreprocess the data by subtracting a bilinear interpolation polynomial,

s = (x− x00)/(x01 − x00), t = (y − y00)/(y01 − y00),

g(x, y) = (1− t)(1 − s)f00 + (1 − t)sf01 + t(1− s)f10 + stf11,

where xij , yij , and fij are the four corner coordinates of the original domain andtheir function values, respectively, before activating the cF shock detection algorithm.The regularity of the boundary can be further increased by using an interpolationpolynomial with a higher degree of smoothness, for example, a bicubic spline.

In the hybrid Compact-WENO scheme, the derivative of the fluxes at bound-ary points are computed by the WENO scheme regardless of the smoothness of theboundary points. So an even extension of the solution does not pose a challenge forthe problems we have tested. In a more general case, one can always require thenumerical flux at the boundary points of the domain be computed by the WENOscheme regardless of its smoothness to avoid this issue at the expense of a slightlyincreased computational time.

Remark 2.7. Since we are mostly interested in solving large scale time dependenthyperbolic PDEs which call for the cF edge detection algorithm at every time step, itis preferable to employ the most efficient CFT algorithm to compute the cF and/orits derivative in order to keep the computational overhead as small as possible.

Remark 2.8. For a periodic problem, similar to a nonperiodic problem, one couldeasily devise the MXM, EOD, and FFT algorithms for computing the cF and/or itsderivative and the remarks discussed above are also generally valid.

2.2. cF shock detection algorithm. We shall define a key parameter Fmean

and a definition that plays a major role in our cF shock detection algorithm.• Knowing the derivative of the cF, we define its mean as

Fmean =1

N + 1

N∑j=0

|fN(xj)|.(2.23)

This is a critical quantity that will be used in partitioning the domain intosubdomains containing one or more discontinuities which will be subjected tofurther detailed analysis.

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• We shall define a set of consecutive grid points in a nonoverlapping subdo-main,

Ωn = {xi| |fN(xi)| > Fmean, i = in, in + 1, . . . , in + s, s > 1},(2.24)

where n = 1, 2, . . . is an integer index of the subdomain.To develop an effective and robust shock detection algorithm, we assume that dis-continuities are singular rare events (finite number of them) with a measure zero andcompactly supported and the function is otherwise sufficiently smooth. As a discon-tinuity forms, increasingly more energy is being accumulated in the high frequencyrange in order to maintain a steady steep slope at the jump locations where thecharacteristic curves are converging. The mean value Fmean gives a measure of theaverage energy of the function being examined, in which the energy contributed bythe discontinuities are being amortized over the whole domain. This separation ofscales allows the algorithm to identify the number of discontinuities as well as thelocation and strength of large jumps in the function with reasonably good accuracyand efficiency, as demonstrated in the examples shown.

We will first describe a general cF shock detection algorithm that works well formost problems containing discontinuities with jumps of similar order of strengths. Insituations where there is a large difference in the jump strength between small andlarge discontinuities, an iterative version of the shock detection algorithm is givenand an example using the Shepp–Logan phantom image is used to demonstrate itsrobustness.

Algorithm 1. cF shock detection algorithm.

1. Compute the derivative of cF fN(xj) and its mean value Fmean.

Use either the MXM, EOD, or CFT algorithms to compute fN with eitherno filtering (σk = 1) or, if needed, the high-pass exponential filter (2.6) toreduce the influence of the low frequency modes.

2. Normalize fN(xj) with its maximum absolute value maxl(fN(xl)).3. Modify the subdomains Ωn into subdomains Ωmod

n .

By comparing fN(xj) with its mean value Fmean,

Ij∗ =

{1, |fN(xj)| > Fmean (jump subdomain),0, otherwise (smooth subdomain).

(2.25)

In the application, a buffer zone of width 2r centered at each j∗ is createdand set them all to 1. The subdomains corresponding to the index belong toIj∗ form Ωmod

n . (In this study, we take r = 3 as 2r − 1 is the order of theWENO scheme.)

4. Generate the WENO Flag at each grid point in the jump subdomain.Within each Ωmod

n , we mark the locations of the minimum and maximum

values of fN as imin and imax, respectively. By defining i0 = min(imin, imax)and i1 = max(imin, imax), the WENO Flag at each grid point will then bedecided accordingly as

Flagi =

{1, i ∈ [i0, i1], nonsmooth WENO stencils,0, otherwise, smooth Compact stencils.

(2.26)

For brevity, the cF shock detection algorithm and MR shock detection algorithmwill be referred to as cF shock detector and MR shock detector, respectively, owingdiscussions.

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cF analysis MR analysis

x

Fu

nctio

n

cF, F

_m

ean

-0.5 0 0.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.5

0

0.5

1

FunctioncFF_mean

x

Fu

nctio

n

MR

, F_

mea

n

-0.5 0 0.5-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

FunctionMRF_mean

Fig. 3. (Left) The derivative of cF fN (xi) with Fmean = 7.2×10−2 and (right) the sixth orderMR coefficients di with Fmean = 9.6× 10−3 for the test function equation (2.27) with N = 256.

Remark 2.9. Optionally, a check for a constant will be performed by examiningthe absolute difference between the minimum and maximum values of the whole datavector, Δfmax = |maxi |fi| − mini |fi||. If Δfmax < ε0 (typically, machine zero ε),we shall assume that no discontinuity exists and the cF shock detector will not beoperated on this data vector.

To demonstrate the performance of the cF shock detector for finding discontinu-ities in a piecewise smooth function and its derivative, we consider the following testfunction,

f(x) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

16 [G(x, β, z − δ) + 4G(x, β, z) +G(x, β, z + δ)], x ∈ [−0.8,−0.6],1, x ∈ [−0.4,−0.2],1− [10(x− 0.1)], x ∈ [0, 0.2],16 [F (x, α, a− δ) + 4F (x, α, a) + F (x, α, a+ δ)], x ∈ [0.4, 0.6],0, else,

G(x, β, z) = e−β(x−z)2, F (x, α, a) =√max(1 − α2(x− a)2, 0),

(2.27)

where z = 0.7, δ = 0.005, β = log 236δ2 , a = 0.5, and α = 10. This function consists of

a smooth Gaussian, a discontinuous Heaviside function, a piecewise linear trianglefunction, and a smooth elliptic function. In Figure 3, we show both the derivativeof cF fN(xi) with mean Fmean = 7.2 × 10−2, and the sixth order MR coefficients diwith mean Fmean = 9.6 × 10−3 for the test function (2.27) with N = 256. Notice

that the strength of the jump of the Heaviside function is well predicted by the cF fN

using the cF shock detector but not by the MR shock detector. From the figure, wecan see that, comparing with the MR analysis, cF analysis gives a very large valueat the discontinuities of the Heaviside function, the corners of the piecewise lineartriangle function, and the smooth elliptic function, along with some oscillations inthe neighborhoods of these shapes. Using the mean Fmean, the cF shock detectorpartitions the domain into several subdomains containing jumps of the function andits derivatives accurately, which will be further analyzed to pinpoint their estimatedjump locations.

2.3. Iterative cF shock detection algorithm. In the example above, wepresented the performance of the cF shock detector for a function with different

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Algorithm 1 (N = 256) Algorithm 2 (N = 256) Algorithm 2 (N = 128)

Fig. 4. The discontinuities Flagy as detected by (Left) the cF shock detector with N = 256;(middle) iterative cF shock detector with N = 256; and (right) N = 128.

forms of shapes. We note that the jumps of different shapes are of similar orderin size and support. Often, the solution, for example, in a high speed compressibleturbulence flow, contains not only both fine smooth structures and discontinuities,but also disparate scales. In this situation, the mean Fmean will still be too large forthe shock detection algorithm to identify the relatively small jump discontinuities.Algorithm 1 described above will fail to detect those small jumps. Thus, an iterativealgorithm is developed to improve the shock detection capability of Algorithm 1 bysubtracting large jumps incrementally from the solution until all relevant jumps havebeen identified. The iterative algorithm described below is very accurate in identifyingmost if not all relevant jump discontinuities subject to certain limitations, however,it is computationally expensive. Thus, the iterative algorithm should only be usedsparingly and when necessary.

Algorithm 2. Iterative cF shock detection algorithm.

1. Compute the cF Ai = fN(xi) and its derivative Bi = fN(xi) of the functionf(x).

2. Search for an index i∗ such that A∗ = |Ai∗ | is the maximum.3. If A∗ > εCF , then

• Set the WENO Flag using the cF shock detector with Bi.• Determine the index j∗ = i∗ − 1 or j∗ = i∗ + 1 such that |Aj∗ | is thenext maximum.• Set the height of the jump to be H∗ = 1

2 ([f ](xi∗ )+ [f ](xj∗))/(1+ εGibbs).• Subtract a Heaviside function H(x, xi∗ , H

∗) with jump H∗ at xi∗ fromf(x).• Repeat the procedure from Step 1 until the maximum number of itera-tions CFmax has reached.

We take the parameters εCF = O(Δx), εGibbs = 0.09, and CFmax = 100.To illustrate the performance of the iterative cF shock detector, we find the

WENO Flag in the x- and y-directions of the Shepp–Logan phantom image, whichis used as a benchmark two-dimensional piecewise smooth data set that models anMRI image of a human brain. It is very challenging for a shock detection algorithmto detect the small structures enclosed inside the head bone as the heights of the headbone range between 1 and 2 while the heights of the small scale soft tissues rangebetween 1.01 and 1.02. The ratio of the jumps from large to small structures is of theorder of hundreds. We applied the iterative cF shock detector on the modified Shepp–Logan phantom image (with a bilinear function added) with resolutions N = 128 andN = 256, and the results are shown in Figure 4. The small scale structures cannot

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be detected even at a relatively high resolution N = 256 using the cF shock detec-tor. However, they are well identified by the iterative cF shock detector even at alower resolution N = 128. Good results had also been obtained by a similar iterativealgorithm based on the radial basis function method [21, 22, 23].

Remark 2.10. In all the examples as well as those not shown in the numericalresults section below, only the cF shock detector is employed.

2.4. The hybrid Compact-WENO finite difference scheme. Now, to-gether with all the numerical algorithms (Compact scheme, WENO scheme, and shockdetectors), we have all the tools to build the hybrid Compact-WENO finite differencescheme. For the details of the hybrid scheme, see [5, 28, 30] and references therein.

Algorithm 3. Hybrid Compact-WENO scheme.

1. Perform the cF or MR analysis on one or more suitable variable(s) (typically,density ρ) once at the beginning of a time stepping scheme.

2. Set a WENO Flag using the shock detection algorithm.3. Create a buffer zone around each grid point xi such that all the grid points

inside the buffer zone are flagged as nonsmooth stencils.This condition prevents the computation of the derivative of the fluxes by theCompact scheme using nonsmooth functional values. If, for example, a gridpoint xi is flagged as a nonsmooth stencil, then its neighboring grid points{xi−m, . . . , xi, . . . , xi+m} will also be designated as nonsmooth stencils, thatis, {Flagj = 1, j = i−m, . . . , i, . . . , i+m} (typically, m = r).

4. Compute the derivative of the fluxes at each cell center by• (nonsmooth WENO subdomain): the WENO scheme.• (smooth Compact subdomain): the Compact scheme.

In other words, the computational domain is subdivided into nonsmooth WENOsubdomains and smooth Compact subdomains based on the shock detection algorithmin which the WENO-Z scheme and the Compact scheme will be applied, respectively.

In the following discussion, we shall denote the hybrid scheme using the cF shockdetector as the Hybrid-cF scheme and the MR shock detector as the Hybrid-MRscheme, and collectively refer to both schemes as the Hybrid scheme. We shall alsodenote Nflag (NCF

flag and NMRflag) as the number of stencils including those within the

buffer zone that are flagged as nonsmooth WENO stencils by a shock detector.

3. Numerical experiments. We consider the two-dimensional Euler equationsfor gas dynamics in strong conservation form:

(3.1) Qt + Fx +Gy = 0,

where the conservative variables Q = (ρ, ρu, ρv, E)T , the fluxes F = (ρu, ρu2 +P, ρuv, (E+P )u)T , andG = (ρv, ρuv, ρv2+P, (E+P )v)T in the x- and y-directions, re-spectively, and the equation of state (EOS) is P = (γ−1)

(E − 1

2ρ(u2 + v2)

), γ = 1.4.

The ρ, u, v, P , and E are the density, velocity in x- and y-directions, pressure, andtotal energy, respectively. Readers are referred to [3, 28, 30] and references containedtherein for details of the spatial schemes and the third order TVD Runge–Kuttatime stepping scheme used in the Hybrid scheme. The CFL condition for stability isCFL = 0.45.

In the following numerical experiments, we present a few preliminary results ob-tained by the Hybrid-cF scheme in solving a system of hyperbolic conservation laws.The density and pressure, along with the Flag showing the stencils using the WENOscheme for computing the fluxes in both directions, of a few classical one- and two-dimensional shocked problems are shown. The CPU timing (in seconds) and the

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t = 2.5 t = 5.0

x

Rh

o

-5 0 5 10 15

1

2

3

4

5

x

Rh

o-5 0 5 10 15

1

2

3

4

5

x-5 0 5 10 15

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

x-5 0 5 10 15

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

tM

ean

0 1 2 3 4 5

0.01

0.02

0.03

Fig. 5. (Top row) The density ρ and WENO Flag (red squares and line) and (bottom row) theabsolute derivative of the cF partial sum of the density of the extended shock-density wave interactionproblem as computed by the Hybrid-cF scheme with N = 800 at times (left) t = 2.5 (NCF

flag = 54)and (middle) t = 5 (NCF

flag = 110), and (right) the temporal evolution of Fmean.

speedup factor of the Hybrid-cF scheme using the MXM, EOD, and CFT algorithmsand the Hybrid-MR scheme over the pure WENO scheme are given showing the effi-cacy of the Hybrid scheme. The one-dimensional extended shock-density wave inter-action problem is simulated with the Hybrid-cF scheme showing the effective spatialand temporal adaptivity of the scheme. The two-dimensional classical Riemann initialvalue problems and the Mach 10 DMR problem are shown to demonstrate that theperformance of the Hybrid scheme can be extended to a higher-dimensional problem.With the exception of the one-dimensional extended shock-density wave interaction,where the γ = 16 order high-pass exponential filter is employed for the cF shockdetector, no filter (that is, σk = 1) is needed in obtaining the results shown. Otherstandard test cases (for example, Lax, Sod, 123, and Woodward–Colella interactingblast wave problems) have been conducted with similar performances metrics. Hence,their results are not shown.

3.1. One-dimensional extended shock-density wave interaction prob-lem. To demonstrate the spatial and temporal adaptivity of the Hybrid-cF scheme,we simulated the one-dimensional Euler equations in an extended domain for a longertime with an initial condition

(ρ, u, P ) =

{( 27

7 ,4√359 , 31

3 ), −5 ≤ x < −4,( 1 + ε sin(kx), 0, 1 ), −4 ≤ x ≤ 15,

where x ∈ [−5, 15], ε = 0.2, and k = 5.The nonlinearity of the Euler equations generates both large fine scale structures

and localized shocklets. The high frequency waves behind the main shock are smoothfunctions but often misidentified as a strong gradient by a lower order shock detectionalgorithm when the solution is slightly underresolved. In Figure 5, the density ρ

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and WENO Flag (red squares and line) as computed by the Hybrid-cF scheme withN = 800 at times t = 2.5 and t = 5, respectively, are shown. The absolute values ofthe derivative of the cF partial sum and its mean Fmean of the density as computedby the Hybrid-cF scheme are also presented at the corresponding times. The resultsshow that the cF shock detector coupled with its Fmean have accurately identifiedthe locations of the main shock and the temporally evolving shocklets, and correctlyclassified the high frequency wave as smooth fine scale structures behind the mainshock. The Hybrid-cF scheme deploys the WENO-Z5 scheme to compute the fluxgradient in an essentially nonoscillatory manner while the high resolution Compactscheme is used for resolving the high frequency waves efficiently. In order to reducethe influence from the large fine scale structures behind the shock on the cF shockdetector, the γ = 16 order high-pass exponential filter is employed in this example.The WENO flag counts are NCF

flag(t = 2.5) = 54 (6%) and NCFflag(t = 5) = 110 (14%).

The temporal evolution of the mean is also given in Figure 5 to show the adaptivityof the cF shock detector in response to the formation of shocklets and high gradients,and the general trend in the solution in time.

3.2. Two-dimensional Riemann initial value problem. To examine theperformance of the Hybrid scheme for higher-dimensional problems, we solve theclassical Riemann initial value problem with initial data in the form

Q = (P, ρ, u, v) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩Q1 = (P1, ρ1, u1, v1) if x > x0 and y ≥ y0,

Q2 = (P2, ρ2, u2, v2) if x ≤ x0 and y ≥ y0,

Q3 = (P3, ρ3, u3, v3) if x ≤ x0 and y ≤ y0,

Q4 = (P4, ρ4, u4, v4) if x > x0 and y < y0.

According to [32], there are 19 genuinely different admissible configurations for poly-tropic gas, separated by the three types of one-dimensional centered waves, namely,

rarefaction (−→R ), shock (

←−S ), and contact wave (J±). The arrows (−→· ) and (←−· ) indi-

cate forward and backward waves, and the superscripts J+ and J− refer to negativeand positive contacts, respectively. We refer readers to [32] for details. We have per-formed calculations on all 19 configurations with the same resolution of 400× 400 asused in [32]. Here, we will only show the three representative results of configurations(3, 5, 12) as computed by the Hybrid-cF scheme. The performance of the Hybrid-cFscheme with other configurations are similar and their results are omitted here.

The density with flooded contours and lines of these three configurations areshown in Figures 6–8, respectively. The large scale structures of the flow agreewell with those in the literature. We remark that the WENO Flag in the x- andy-directions are very sharp with only a few grid points contained in each segmentshowing the accuracy of the cF shock detector. Table 1 presents the CPU timingsalong with the speedup factor in the simulation of the two-dimensional Riemann ini-tial value problem with configuration 3 computed by the pure WENO-Z5, Hybrid-cF(including MXM, EOD, and CFT algorithms), and Hybrid-MR schemes. From theresults, we can see that the Hybrid schemes are substantially faster than the pureWENO-Z5 scheme with speedup factor increased from 1.5 to 3.0 as the resolution in-creased. The cF shock detector with the CFT algorithm has a slight speed advantageover the EOD algorithm at high resolutions and is a definite win over the MXM algo-rithm. The Hybrid-MR scheme is always slightly faster than the Hybrid-cF schemeas the amount of work required for detecting discontinuities is O(N) and O(N logN),respectively. Methods to further reduce the CPU time required for the cF shock de-

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Configuration 3 : (←−−R21,

←−−R32,

←−−R34,

←−−R41), (x0, y0) = (0.8, 0.8), tf = 0.8.

Q =

⎧⎪⎪⎨⎪⎪⎩( 1.5, 1.5, 0, 0 )( 0.3, 0.5323, 1.206, 0 )( 0.029, 0.138, 1.206, 1.206 )( 0.3, 0.5323, 0, 1.206 )

.

Density ρ Pressure P

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

11.641.521.301.181.060.930.810.690.570.440.320.20

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.551.411.291.130.980.820.670.510.360.200.05

Flagx Flagy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 6. Two-dimensional Riemann initial value problem with configuration 3: density ρ, pres-sure P , Flagx and Flagy in x- and y-directions, respectively.

tector, such as restricting the shock detection algorithm to be activated only aroundpotential discontinuities, is under study and will be reported in the future.

3.3. Two-dimensional Mach 10 DMR problem. To illustrate the efficiencyof the Hybrid scheme in a more practical problem, we applied both Hybrid schemesto solve the two-dimensional Mach 10 DMR problem. The problem and its numericalresults are well documented in the literature [37].

We run both the Hybrid-cF and Hybrid-MR schemes with resolution 800 × 200uniform cells to a final time tf = 0.2. The density flooded contours and lines areshown in Figure 9. The large scale structures (for example, the triple point, the inci-dent shock, the reflected shock, a Mach stem, and a slip plane) of the flow agree verywell with each other and with those in the literature. In Figure 10, the small scalestructures (for example, the small vortical rollups along the slip line and the large

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Configuration 5 : (J−21, J

−32, J

−34, J

−41), (x0, y0) = (0.5, 0.5), tf = 0.23.

Q =

⎧⎪⎪⎨⎪⎪⎩( 1, 1, −0.75, −0.5 )( 1, 2, −0.75, 0.5 )( 1, 1, 0.75, 0.5 )( 1, 3, 0.75, −0.5 )

.


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

13.803.543.282.892.632.372.111.781.591.331.07

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

12.202.061.921.781.641.511.371.231.090.920.780.640.50

Flagx Flagy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 7. Two-dimensional Riemann initial value problem with configuration 5: density ρ, pres-sure P , Flagx and Flagy in x- and y-directions, respectively.

mushroom shaped vortical rollup at the tip of the jet) around the region x ∈ [2.15, 2.9]behind the incident shock are shown. Besides some expected minor discrepancies inthe small scale vortical rollups along the slip line and the jet at high resolution, thefine scale structures agree very well with each other between the Hybrid schemesat the given resolution. The WENO Flag in the x- and y-directions of the Hybridschemes are shown in Figure 11. Generally speaking, the Hybrid-cF scheme capturesthe high gradients more accurately and sharply than the Hybrid-MR scheme.

4. Conclusion. We have developed the Hybrid Compact-WENO finite differ-ence scheme with the cF shock detection algorithm (Hybrid-cF) for solving hyperbolicconservation laws with a discontinuous solution in a nonperiodic Cartesian domain.We utilize the derivative of the cF partial sum and its mean to partition the phys-ical domain robustly into nonsmooth and smooth subdomains. The shock locationsare then identified and flagged for special treatment using the WENO scheme. The

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Configuration 12 : (−→S21, J

+32, J

+34,−→S41), (x0, y0) = (0.5, 0.5), tf = 0.25.

Q =

⎧⎪⎪⎨⎪⎪⎩( 0.4, 0.5313, 0, 0 )( 1, 1, 0.7276, 0 )( 1, 0.8, 0, 0 )( 1, 1, 0, 0.7276 )

.


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

11.641.551.461.371.281.191.101.010.910.820.720.630.54

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

12.121.981.841.701.561.421.281.141.010.870.730.590.45

Flagx Flagy

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Fig. 8. Two-dimensional Riemann initial value problem with configuration 12: density ρ,pressure P , Flagx and Flagy in x- and y-directions, respectively.

Table 1

The CPU timings (in seconds) and speedup factors r of two-dimensional Riemann initial valueproblem with configuration 3 computed by the pure WENO-Z5, Hybrid-cF, and Hybrid-MR schemeswith εMR = 5× 10−3.

N ×M WENO-Z5Hybrid

Hybrid-cF Hybrid-MRMXM r EOD r CFT r MR r

128 × 128 57 41 1.4 38 1.5 38 1.5 42 1.4256 × 256 489 262 1.9 250 2.0 245 2.0 235 2.1512 × 512 4426 2065 2.1 1895 2.3 1754 2.5 1529 2.9

MXM, EOD, and CFT algorithms for computing the discrete cF partial sum and itsderivative are derived. We have addressed the respective advantages and disadvan-tages in their implementations, usages, and speeds along with other technical issues.

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Hybrid-cF Hybrid-MR

0 0.5 1 1.5 2 2.5 30

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1

0 0.5 1 1.5 2 2.5 30

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1

0 0.5 1 1.5 2 2.5 30

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1

0 0.5 1 1.5 2 2.5 30

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1

Fig. 9. The density contour flood and lines of the Mach 10 double Mach shock reflection problemas computed by (left) the Hybrid-cF scheme and (right) the Hybrid-MR scheme at time tf = 0.2.

Hybrid-cF Hybrid-MR

2.2 2.4 2.6 2.80

0.1

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0.4

0.5

2.2 2.4 2.6 2.80

0.1

0.2

0.3

0.4

0.5

Fig. 10. The zoom in of the density of the Mach 10 double Mach shock reflection problem ascomputed by (left) the Hybrid-cF scheme and (right) the Hybrid-MR scheme at time tf = 0.2.

Hybrid-cF Hybrid-MR

WENO

Flagin

x

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 30

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0.4

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1

WENO

Flagin

y

0 0.5 1 1.5 2 2.5 30

0.2

0.4

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0.8

1

0 0.5 1 1.5 2 2.5 30

0.2

0.4

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1

Fig. 11. Distribution of the WENO flags Flagx and Flagy in the x- and y-directions, respec-tively, of the Hybrid-cF and Hybrid-MR schemes at time tf = 0.2.

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We then present the cF shock detector and its iterative version for problems withjumps of large differences in scales. The iterative cF shock detector is able to identifyedges of the small scale structures in the Shepp–Logan phantom image clearly evenat a lower resolution.

The Hybrid-cF scheme is then applied to one- and two-dimensional hyperbolicconservation laws. The simulation of the one-dimensional extended shock-densitywave interaction problem and two-dimensional classical Riemann initial valve prob-lems demonstrate the spatial and temporal adaptivity of the Hybrid-cF scheme insolving a problem containing strong shock, multiple developing shocklets, and highfrequency waves behind the strong shock. The two-dimensional Mach 10 DMR prob-lem is also simulated to show that the Hybrid-cF scheme works as well as the Hybrid-MR scheme. These preliminary results are in good agreement with those obtainedin the literature and can potentially be very useful for certain classes of shockedflows containing both discontinuities and small scale structures, such as high speedcompressible turbulent flows. Furthermore, a speedup of up to 2 or more in CPUtimings is achieved with the Hybrid-cF (CFT) scheme when compared with the pureWENO-Z5 scheme.

In our future work, we plan to use the high order shock detectors to classifydiscontinuities of different types, such as contact waves, shocks, rarefaction waves, andmaterial interfaces, and to apply appropriate numerical techniques to obtain the bestpossible solution. We are also planning to apply the Hybrid-cF scheme for simulatingcompressible turbulence and chemically reactive multicomponents multiphase flows,and to study the related theoretical and practical issues (nonuniform, finite volume orfinite element grids, parallelization), and to explore alternative advanced techniquesin identifying discontinuities.

Acknowledgments. We appreciate the careful reading and valuable suggestionsmade to the contents of the manuscript by the reviewers and Dr. Oleg Schilling ofLawrence Livermore National Laboratory. The authors would also like to thank QingCheng of Xiamen University for the preliminary work done on the conjugate Fouriershock detection method during the summer workshop in Advanced Research in Ap-plied Mathematics and Scientific Computing 2014 at the School of MathematicalSciences of Ocean University of China.

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