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24
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...
Balancing Source Terms and Flux Gradients in HighResolution Godunov Methods: The QuasiSteady WavePropogation Algorithm
 J. Comput. Phys
, 1998
"... . Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of suc ..."
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Cited by 54 (5 self)
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. Conservation laws with source terms often have steady states in which the flux gradients are nonzero but exactly balanced by source terms. Many numerical methods (e.g., fractional step methods) have difficulty preserving such steady states and cannot accurately calculate small perturbations of such states. Here a variant of the wavepropagation algorithm is developed which addresses this problem by introducing a Riemann problem in the center of each grid cell whose flux difference exactly cancels the source term. This leads to modified Riemann problems at the cell edges in which the jump now corresponds to perturbations from the steady state. Computing waves and limiters based on the solution to these Riemann problems gives highresolution results. The 1D and 2D shallow water equations for flow over arbitrary bottom topography are use as an example, though the ideas apply to many other systems. The method is easily implemented in the software package clawpack. Keywords: Godunov meth...
A WellBalanced Scheme Using NonConservative Products Designed for Hyperbolic Systems of Conservation Laws With Source Terms
, 2001
"... The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a nonconservative reformulation of the zeroorder terms of the righthandside of the equations. In this context, we decided to use the ..."
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Cited by 21 (3 self)
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The aim of this paper is to present a new kind of numerical processing for hyperbolic systems of conservation laws with source terms. This is achieved by means of a nonconservative reformulation of the zeroorder terms of the righthandside of the equations. In this context, we decided to use the results of DalMaso, LeFloch and Murat [9] about nonconservative products, and the generalized Roe matrixes introduced by Toumi [36] to derive a firstorder linearized wellbalanced scheme in the sense of Greenberg and LeRoux [19]. As a main feature, this approach is able to preserve the right asymptotic behaviour of the original inhomogeneous system [31], which is not a obvious property [6]. Numerical results for the Euler equations are shown to handle correctly these equilibria in various situations. Key words: conservation laws, source terms. nonconservative products, balanced scheme. AMS subjects classification: 65M06, 76N15. 1 Current adress: Foundation for Research and Technology Hel...
Capturing Shock Reflections: An improved flux formula
 J. Comput. Phys
, 1996
"... Godunov type schemes, based on exact or approximate solutions to the Riemann problem, have proven to be an excellent tool to compute approximate solutions to hyperbolic systems of conservation laws. However, there are many instances in which a particular scheme produces inappropriate results. In thi ..."
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Cited by 18 (5 self)
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Godunov type schemes, based on exact or approximate solutions to the Riemann problem, have proven to be an excellent tool to compute approximate solutions to hyperbolic systems of conservation laws. However, there are many instances in which a particular scheme produces inappropriate results. In this paper we consider several situations in which Roe's scheme gives incorrect results (or blows up all together) and propose an alternative flux formula that produces numerical approximations in which the pathological behavior is either eliminated or reduced to computationally acceptable levels. Key Words. Nonlinear Systems of Conservation Laws, Shock Capturing schemes, Shock Reflections. AMSMOS Classification: Primary 65M05, Secondary 65M10 1 Introduction Shock capturing techniques for the computation of discontinuous solutions to hyperbolic conservation laws are based on an old (by now) theorem of Lax and Wendroff establishing that the limit solutions of a consistent scheme in conservat...
Localization effects and measure source terms in numerical schemes for balance laws
 Math. Comp
"... Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and ..."
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Cited by 13 (3 self)
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Abstract. This paper investigates the behavior of numerical schemes for nonlinear conservation laws with source terms. We concentrate on two significant examples: relaxation approximations and genuinely nonhomogeneous scalar laws. The main tool in our analysis is the extensive use of weak limits and nonconservative products which allow us to describe accurately the operations achieved in practice when using Riemannbased numerical schemes. Some illustrative and relevant computational results are provided. 1.
On convergence of numerical schemes for hyperbolic conservation laws with stiff source terms
 Math. Comp
, 1997
"... Abstract. We deal in this study with the convergence of a class of numerical schemes for scalar conservation laws including stiff source terms. We suppose that the source term is dissipative but it is not necessarily a Lipschitzian function. The convergence of the approximate solution towards the en ..."
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Cited by 12 (1 self)
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Abstract. We deal in this study with the convergence of a class of numerical schemes for scalar conservation laws including stiff source terms. We suppose that the source term is dissipative but it is not necessarily a Lipschitzian function. The convergence of the approximate solution towards the entropy solution is established for first and second order accurate MUSCL and for splitting semiimplicit methods. 1.
Upwind and HighResolution Methods for Compressible Flows: From Donor . . .
, 2006
"... In this paper I review three key topics in CFD that have kept researchers busy for half a century. First, the concept of upwind differencing, evident for 1D linear advection. Second, its implementation for nonlinear systems in the form of highresolution schemes, now regarded as classical. Third, it ..."
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Cited by 11 (2 self)
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In this paper I review three key topics in CFD that have kept researchers busy for half a century. First, the concept of upwind differencing, evident for 1D linear advection. Second, its implementation for nonlinear systems in the form of highresolution schemes, now regarded as classical. Third, its genuinely multidimensional implementation in the form of residualdistribution schemes, the most recent addition. This lecture focuses on historical developments; it is not intended as a technical review of methods, hence the lack of formulas and absence of figures.
Diffusion Limit Of The Lorentz Model: Asymptotic Preserving Schemes
"... This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right ..."
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Cited by 10 (2 self)
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This paper deals with the diusion limit of a kinetic equation where the collisions are modeled by a Lorentz type operator. The main aim is to construct a discrete scheme to approximate this equation which gives for any value of the Knudsen number, and in particular at the diusive limit, the right discrete diusion equation with the same value of the diusion coecient as in the continuous case. We are also naturally interested with a discretization which can be used with few velocity discretization points, in order to reduce the cost of computation.
Approximate Projection Methods for Incompressible Flow: Implementation, Variants and Robustness.
 LANL UNCLASSIFIED REPORT LAUR942000, LOS ALAMOS NATIONAL LABORATORY
, 1995
"... We present an effective method for computing accurate solutions to timedependent incompressible fluidflow problems. The method uses an approximate projection method coupled with highorder Godunov convection, and a CrankNicholson method for diffusive terms. Linear systems of equations are com ..."
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Cited by 9 (0 self)
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We present an effective method for computing accurate solutions to timedependent incompressible fluidflow problems. The method uses an approximate projection method coupled with highorder Godunov convection, and a CrankNicholson method for diffusive terms. Linear systems of equations are computed with a multigrid algorithm. The algorithm is demonstrated for incompressible, Boussinesq, and incompressible variable density flows. This algorithm does particularly well at computing high Reynolds number, and inviscid flows as well as those with discontinuous data. We
SecondOrder Characteristic Methods for AdvectionDiffusion Equations and Comparison to Other Schemes
 Advances in Water Resources
, 1999
"... We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order RungeKutta approximation of the characteristics within the framework of the EulerianLagrangian localized adjoint method. These methods naturally incorporate all three type ..."
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Cited by 4 (1 self)
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We develop two characteristic methods for the solution of the linear advection diffusion equations which use a second order RungeKutta approximation of the characteristics within the framework of the EulerianLagrangian localized adjoint method. These methods naturally incorporate all three types of boundary conditions in their formulations, are fully mass conservative, and generate regularly structured systems which are symmetric and positive definite for most combinations of the boundary conditions. Extensive numerical experiments are presented which compare the performance of these two RungeKutta methods to many other well perceived and widely used methods which include many Galerkin methods and high resolution methods from #uid dynamics. Key words characteristic methods, comparison of numerical methods, EulerianLagrangian methods, numerical solutions of advectiondi#usion equations, RungeKutta methods. 1 Introduction Advectiondi#usion equations are an important cla...