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136
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 158 (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
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...
Analysis and Design of Numerical Schemes for Gas Dynamics 1 Artificial Diffusion, Upwind Biasing, Limiters and Their Effect on Accuracy and Multigrid Convergence
 INTERNATIONAL JOURNAL OF COMPUTATIONAL FLUID DYNAMICS
, 1995
"... The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, w ..."
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Cited by 77 (40 self)
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The theory of nonoscillatory scalar schemes is developed in this paper in terms of the local extremum dimin ishing (LED) principle that maxima should not increase and minima should not decrease. This principle can be used for multidimensional problems on both structured and unstructured meshes, while it is equivalent to the total variation diminishing (TVD) principle for onedimensional problems. A new formulation of symmetric limited positive (SLIP) schemes is presented, which can be generalized to produce schemes with arbitrary high order of accuracy in regions where the solution contains no extrema, and which can also be implemented on multidimensional unstructured meshes. Systems of equations lead to waves traveling with distinct speeds and possibly in opposite directions. Alternative treatments using characteristic splitting and scalar diffusive fluxes are examined, together with a modification of the scalar diffusion through the addition of pressure differences to the momentum equations to produce full upwinding in supersonic flow. This convective upwind and split pressure (CUSP) scheme exhibits very rapid convergence in multigrid calculations of transonic flow, and provides excellent shock resolution at very high Mach numbers.
A Massively Parallel Adaptive Finite Element Method with Dynamic Load Balancing
 Appl. Numer. Math
, 1993
"... We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta meth ..."
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Cited by 70 (13 self)
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We construct massively parallel adaptive finite element methods for the solution of hyperbolic conservation laws. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a RungeKutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We demonstrate parallel efficiency through computations on a 1024processor nCUBE/2 hypercube. We present results using adaptiverefinement to reduce the computational cost of the method, and tiling, a dynamic, elementbased data migration system that maintains global load balance of the adaptive method by overlapping neighborhoods of processors that each perform local balancing. 1. Introduction We are studying massively parallel adaptive finite element methods for solving systems ofdimensional hyper...
Wave Propagation Algorithms for Multidimensional Hyperbolic Systems
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
"... ..."
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 60 (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.
Total variation diminishing RungeKutta schemes
 Math. Comp
, 1998
"... In this paper we further explore a class of high order TVD �total variation diminishing� RungeKutta time discretization initialized in �12�, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that nonTVD but linearly stable ..."
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Cited by 60 (7 self)
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In this paper we further explore a class of high order TVD �total variation diminishing� RungeKutta time discretization initialized in �12�, suitable for solving hyperbolic conservation laws with stable spatial discretizations. We illustrate with numerical examples that nonTVD but linearly stable RungeKutta time discretization can generate oscillations even for TVD �total variation diminishing � spatial discretization, verifying the claim that TVD RungeKutta methods are important for such applications. We then explore the issue of optimal TVD RungeKutta methods for second, third and fourth order, and for low storage RungeKutta methods. 1 Supported by a ARPANDSEG graduate student fellowship.
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...
Robust Numerical Methods for PDE Models of Asian Options
 Journal of Computational Finance
, 1998
"... We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this p ..."
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Cited by 46 (14 self)
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We explore the pricing of Asian options by numerically solving the the associated partial differential equations. We demonstrate that numerical PDE techniques commonly used in finance for standard options are inaccurate in the case of Asian options and illustrate modifications which alleviate this problem. In particular, the usual methods generally produce solutions containing spurious oscillations. We adapt flux limiting techniques originally developed in the field of computational fluid dynamics in order to rapidly obtain accurate solutions. We show that flux limiting methods are total variation diminishing (and hence free of spurious oscillations) for nonconservative PDEs such as those typically encountered in finance, for fully explicit, and fully and partially implicit schemes. We also modify the van Leer flux limiter so that the secondorder total variation diminishing property is preserved for nonuniform grid spacing. 1 Introduction Asian options are securities with payoffs...