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178
The Relaxation Schemes for Systems of Conservation Laws in Arbitrary Space Dimensions
 Comm. Pure Appl. Math
, 1995
"... We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with ..."
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Cited by 166 (20 self)
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We present a class of numerical schemes (called the relaxation schemes) for systems of conservation laws in several space dimensions. The idea is to use a local relaxation approximation. We construct a linear hyperbolic system with a stiff lower order term that approximates the original system with a small dissipative correction. The new system can be solved by underresolved stable numerical discretizations without using either Riemann solvers spatially or a nonlinear system of algebraic equations solver temporally. Numerical results for 1D and 2D problems are presented. The second order schemes are shown to be total variation diminishing (TVD) in the zero relaxation limit for scalar equations. 1. Introduction In this paper we present a class of nonoscillatory numerical schemes for systems of conservation laws in several space dimensions. The basic idea is to use a local relaxation approximation. For any given system of conservation laws, we will construct a corresponding linear hyp...
Essentially nonoscillatory and weighted essentially nonoscillatory schemes for hyperbolic conservation laws
, 1998
"... In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di ere ..."
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Cited by 137 (18 self)
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In these lecture notes we describe the construction, analysis, and application of ENO (Essentially NonOscillatory) and WENO (Weighted Essentially NonOscillatory) schemes for hyperbolic conservation laws and related HamiltonJacobi equations. ENO and WENO schemes are high order accurate nite di erence schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics. These lecture notes are basically selfcontained. It is our hope that with these notes and with the help of the quoted references, the readers can understand the algorithms and code
New HighResolution Central Schemes for Nonlinear Conservation Laws and ConvectionDiffusion Equations
 J. Comput. Phys
, 2000
"... this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution ..."
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Cited by 115 (15 self)
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this paper we introduce a new family of central schemes which retain the simplicity of being independent of the eigenstructure of the problem, yet which enjoy a much smaller numerical viscosity (of the corresponding order )).In particular, our new central schemes maintain their highresolution independent of O(1/#t ), and letting #t 0, they admit a particularly simple semidiscrete formulation
Nonoscillatory Central Schemes For Multidimensional Hyperbolic Conservation Laws
 SIAM J. Sci. Comput
, 1998
"... We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory ..."
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Cited by 99 (14 self)
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We construct, analyze, and implement a new nonoscillatory highresolution scheme for twodimensional hyperbolic conservation laws. The scheme is a predictorcorrector method which consists of two steps: starting with given cell averages, we first predict pointvalues which are based on nonoscillatory piecewiselinear reconstructions from the given cell averages; at the second corrector step, we use staggered averaging, together with the predicted midvalues, to realize the evolution of these averages. This results in a secondorder, nonoscillatory central scheme, a natural extension of the onedimensional secondorder central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408448]. As in the onedimensional case, the main feature of our twodimensional scheme is simplicity. In particular, this central scheme does not require the intricate and timeconsuming (approximate) Riemann solvers which are essential for the highresolution upwind schemes; in fact, even the com...
Strong StabilityPreserving HighOrder Time Discretization Methods
 SIAM Rev
, 2001
"... . In this paper wer#1A43 andfur#41; develop a class ofstr#1X stabilitypr#it.7A3A (SSP) highor#/3 time discr#:44.2XFFX for semidiscr#2X method of lines appr# ximations ofpar# tialdi#er#4 tial equations.Pr#;31.r#X ter#at TVD (total var#X1F1. diminishing) time discr#3:/.2XX;31 thesehighor#A3 timedis ..."
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Cited by 63 (10 self)
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. In this paper wer#1A43 andfur#41; develop a class ofstr#1X stabilitypr#it.7A3A (SSP) highor#/3 time discr#:44.2XFFX for semidiscr#2X method of lines appr# ximations ofpar# tialdi#er#4 tial equations.Pr#;31.r#X ter#at TVD (total var#X1F1. diminishing) time discr#3:/.2XX;31 thesehighor#A3 timediscr#/A3.2X7; methodspr#77A: e thestr#7: stabilitypr #1 er#1/; offir#A3;;.2X Euler time stepping and havepr# ved ver# useful, especially in solving hyper# olicpar#.1: di#er#: tial equations.The new developments in this paper include theconstr#X.2X3 of optimal explicit SSPlinear RungeKutta methods,their application to thestr#1F stability of coer#74 eappr# ximations, a systematic study of explicit SSP multistep methodsfor nonlinear pr#linear and the study of the SSP pr#. er# y of implicit RungeKutta and multistep methods. Key words.str#14 stabilitypr#1XX.27F/ RungeKutta methods, multistep methods, highor#.2 accur #cu , timediscr#43;.27F3 AMS subjectclctj44k7kj3,N 65M20, 65L06 PII. S003614450036757X 1.
Third Order Nonoscillatory Central Scheme For Hyperbolic Conservation Laws
"... . A thirdorder accurate Godunovtype scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: #1. A nonoscillatory piecewisequadratic reconstruction of pointvalues from their given cell averages; and #2. A central differencing ..."
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Cited by 58 (15 self)
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. A thirdorder accurate Godunovtype scheme for the approximate solution of hyperbolic systems of conservation laws is presented. Its two main ingredients include: #1. A nonoscillatory piecewisequadratic reconstruction of pointvalues from their given cell averages; and #2. A central differencing based on staggered evolution of the reconstructed cell averages. This results in a thirdorder central scheme, an extension along the lines of the secondorder central scheme of Nessyahu and Tadmor [NT]. The scalar scheme is nonoscillatory (and hence  convergent), in the sense that it does not increase the number of initial extrema (as does the exact entropy solution operator). Extension to systems is carried out by componentwise application of the scalar framework. In particular, we have the advantage that, unlike upwind schemes, no (approximate) Riemann solvers, fieldbyfield characteristic decompositions, etc., are required. Numerical experiments confirm the highresolution content of...
Numerical Schemes For Hyperbolic Conservation Laws With Stiff Relaxation Terms
 J. Comput. Phys
, 1996
"... Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution meth ..."
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Cited by 56 (11 self)
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Hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a longtime behavior governed by reduced systems that are parabolic in nature. In this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid. We introduce a modification of higher order Godunov methods that possesses the correct asymptotic behavior, allowing the use of coarse grids (large cell Peclet numbers). The idea is to build into the numerical scheme the asymptotic balances that lead to this behavior. Numerical experiments on 2 \Theta 2 systems verify our analysis. 1 Email address: jin@math.gatech.edu 2 Email address: lvrmr@math.arizona.edu Typeset by A M ST E X 2 1. Introduction Hyperbolic systems of partial differential equations that arise in applications ofter have re...
SemiDiscrete CentralUpwind Schemes for Hyperbolic Conservation Laws and HamiltonJacobi Equations
 SIAM J. Sci. Comput
, 2000
"... We introduce new Godunovtype semidiscrete central schemes for hyperbolic systems of conservation laws and HamiltonJacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation, and can be viewed as a generalization of the schemes from [26, 24 ..."
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Cited by 52 (5 self)
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We introduce new Godunovtype semidiscrete central schemes for hyperbolic systems of conservation laws and HamiltonJacobi equations. The schemes are based on the use of more precise information about the local speeds of propagation, and can be viewed as a generalization of the schemes from [26, 24, 25] and [27]. The main advantages of the proposed central schemes are the high resolution, due to the smaller amount of the numerical dissipation, and the simplicity. There are no Riemann solvers and characteristic decomposition involved, and this makes them a universal tool for a wide variety of applications. At the same time, the developed schemes have an upwind nature, since they respect the directions of wave propagation by measuring the onesided local speeds. This is the reason why we call them centralupwind schemes. The constructed schemes are applied to various problems, such as the Euler equations of gas dynamics, the HamiltonJacobi equations with convex and nonconvex Hamiltoni...
HighResolution Nonoscillatory Central Schemes With Nonstaggered Grids For Hyperbolic Conservation Laws
 SIAM J. Numer. Anal
, 1998
"... We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recent ..."
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Cited by 41 (14 self)
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We present a general procedure to convert schemes which are based on staggered spatial grids into nonstaggered schemes. This procedure is then used to construct a new family of nonstaggered, central schemes for hyperbolic conservation laws by converting the family of staggered central schemes recently introduced in [H. Nessyahu and E. Tadmor, J. Comput. Phys., 87 (1990), pp. 408463; X. D. Liu and E. Tadmor, Numer. Math., 79 (1998), pp. 397425; G. S. Jiang and E. Tadmor, SIAM J. Sci. Comput., 19 (1998), pp. 18921917]. These new nonstaggered central schemes retain the desirable properties of simplicity and high resolution, and in particular, they yield Riemannsolverfree recipes which avoid dimensional splitting. Most important, the new central schemes avoid staggered grids and hence are simpler to implement in frameworks which involve complex geometries and boundary conditions.
Compact central WENO schemes for multidimensional conservation laws
 SIAM J. Sci. Comput
, 2000
"... We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greate ..."
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Cited by 38 (10 self)
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We present new third and fifthorder Godunovtype central schemes for approximating solutions of the HamiltonJacobi (HJ) equation in an arbitrary number of space dimensions. These are the first central schemes for approximating solutions of the HJ equations with an order of accuracy that is greater than two. In two space dimensions we present two versions for the thirdorder scheme: one scheme that is based on a genuinely twodimensional Central WENO reconstruction, and another scheme that is based on a simpler dimensionbydimension reconstruction. The simpler dimensionbydimension variant is then extended to a multidimensional fifthorder scheme. Our numerical examples in one, two and three space dimensions verify the expected order of accuracy of the schemes. Key words. HamiltonJacobi equations, central schemes, high order, WENO, CWENO.