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128
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 160 (19 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 125 (17 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 100 (14 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 80 (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 61 (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.
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 57 (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...
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 47 (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...
Approximate Solutions of Nonlinear Conservation Laws and Related Equations
, 1997
"... During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical ..."
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Cited by 34 (11 self)
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During the recent decades there was an enormous amount of activity related to the construction and analysis of modern algorithms for the approximate solution of nonlinear hyperbolic conservation laws and related problems. To present some aspects of this successful activity, we discuss the analytical tools which are used in the development of convergence theories for these algorithms. These include classical compactness arguments (based on BV a priori estimates), the use of compensated compactness arguments (based on H^1compact entropy production), measure valued solutions (measured by their negative entropy production), and finally, we highlight the most recent addition to this bag of analytical tools  the use of averaging lemmas which yield new compactness and regularity results for nonlinear conservation laws and related equations. We demonstrate how these analytical tools are used in the convergence analysis of approximate solutions for hyperbolic conservation laws and related equations. Our discussion includes examples of Total Variation Diminishing (TVD) finitedifference schemes; error estimates derived from the onesided stability of Godunovtype methods for convex conservation laws (and their multidimensional analogue  viscosity solutions of demiconcave HamiltonJacobi equations); we outline, in the onedimensional case, the convergence proof of finiteelement streamlinediffusion and spectral viscosity schemes based on the divcurl lemma; we also address the questions of convergence and error estimates for multidimensional finitevolume schemes on nonrectangular grids; and finally, we indicate the convergence of approximate solutions with underlying kinetic formulation, e.g., finitevolume and relaxation schemes, once their regularizing effect is quantified in terms of the averaging lemma.
Multiphase Computations in Geometrical Optics
 J. Comp. Appl. Math
, 1996
"... In this work we propose a new set of partial differential equations (PDEs) which can be seen as a generalization of the classical eikonal and transport equations, to allow for solutions with multiple phases. The traditional geometrical optics pair of equations suffer from the fact that the class of ..."
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Cited by 32 (1 self)
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In this work we propose a new set of partial differential equations (PDEs) which can be seen as a generalization of the classical eikonal and transport equations, to allow for solutions with multiple phases. The traditional geometrical optics pair of equations suffer from the fact that the class of physically relevant solutions is limited. In particular, it does not include solutions with multiple phases, corresponding to crossing waves. Our objective has been to generalize these equations to accommodate solutions containing more than one phase. The new equations are based on the same high frequency approximation of the scalar wave equation as the eikonal and the transport equations. However, they also incorporate a finite superposition principle. The maximum allowed number of intersecting waves in the solution can be chosen arbitrarily, but a higher number means that a larger system of PDEs must be solved. The PDEs form a hyperbolic system of conservation laws with source terms. Altho...
Central WENO Schemes for Hyperbolic Systems of Conservation Laws
 MATH. MODEL. NUMER. ANAL
, 2001
"... We present a family of highorder, essentially nonoscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially NonOscillatory (WENO) reconstruction of pointvalues from cellaverages ..."
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Cited by 31 (12 self)
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We present a family of highorder, essentially nonoscillatory, central schemes for approximating solutions of hyperbolic systems of conservation laws. These schemes are based on a new centered version of the Weighed Essentially NonOscillatory (WENO) reconstruction of pointvalues from cellaverages, which is then followed by an accurate approximation of the fluxes via a natural continuous extension of RungeKutta solvers. We explicitly construct the third and fourthorder scheme and demonstrate their highresolution properties in several numerical tests.