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56
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...
Hyperbolic Conservation Laws with Stiff Relaxation Terms and Entropy
 Comm. Pure Appl. Math
, 1992
"... We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N \Theta N systems that ..."
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Cited by 127 (8 self)
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We study the limiting behavior of systems of hyperbolic conservation laws with stiff relaxation terms. Reduced systems, inviscid and viscous local conservation laws, and weakly nonlinear limits are derived through asymptotic expansions. An entropy condition is introduced for N \Theta N systems that ensures the hyperbolicity of the reduced inviscid system. The resulting characteristic speeds are shown to be interlaced with those of the original system. Moreover, the first correction to the reduced system is shown to be dissipative. A partial converse is proved for 2 \Theta 2 systems. This structure is then applied to study the convergence to the reduced dynamics for the 2 \Theta 2 case. 1 Email address: cheng@zaphod.uchicago.edu 2 Email address: lvrmr@math.arizona.edu 3 Email address: liu@pde.stanford.edu 2 1. Introduction We are concerned with the phenomena of relaxation, particularly the question of stability and singular limits of zero relaxation time. Relaxation is import...
Efficient AsymptoticPreserving (AP) Schemes For Some Multiscale Kinetic Equations
 SIAM J. Sci. Comp
, 1999
"... . Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the NavierStokes type parabolic equations as the small scaling parameter approaches zero. In practical applications, it is desirable to develop a class of numerical schemes that can work uniformly with respect to ..."
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Cited by 59 (17 self)
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. Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the NavierStokes type parabolic equations as the small scaling parameter approaches zero. In practical applications, it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter, from the rarefied kinetic regimes to the hydrodynamic diffusive regimes. An earlier approach in [11] reformulates such systems into the common hyperbolic relaxation system by Jin and Xin for hyperbolic conservation laws used to construct the relaxation schemes, and then use a multistep time splitting method to solve the relaxation system. Here we observe that the combination of the two timesplit steps may yield a hyperbolicparabolic systems that are more advantageous, in both stability and efficiency, for numerical computations. We show that such an approach yields a class of asymptoticpreserving (AP) schemes which are suitable for the computation of multiscale...
ImplicitExplicit RungeKutta schemes and applications to hyperbolic systems with relaxation
 Journal of Scientific Computing
, 2000
"... We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The sch ..."
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Cited by 43 (6 self)
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We consider implicitexplicit (IMEX) Runge Kutta methods for hyperbolic systems of conservation laws with stiff relaxation terms. The explicit part is treated by a strongstabilitypreserving (SSP) scheme, and the implicit part is treated by an Lstable diagonally implicit Runge Kutta (DIRK). The schemes proposed are asymptotic preserving (AP) in the zero relaxation limit. High accuracy in space is obtained by finite difference discretization with Weighted Essentially Non Oscillatory (WENO) reconstruction. After a brief description of the mathematical properties of the schemes, several applications will be presented. Keywords: RungeKutta methods, hyperbolic systems with relaxation, stiff systems, high order shock capturing schemes. AMS Subject Classification: 65C20, 82D25 1
Convergence of Relaxation Schemes for Conservation Laws
, 1996
"... . We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin, we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and se ..."
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Cited by 33 (10 self)
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. We study the stability and the convergence for a class of relaxing numerical schemes for conservation laws. Following the approach recently proposed by S. Jin and Z. Xin, we use a semilinear local relaxation approximation, with a stiff lower order term, and we construct some numerical first and second order accurate algorithms, which are uniformly bounded in the L 1 and BV norms with respect to the relaxation parameter. The relaxation limit is also investigated. Key words and phrases: Relaxation schemes, conservation laws, shock waves, entropy conditions, hyperbolic singular perturbations. 1. Introduction In this paper we investigate a new class of numerical schemes, which are based on the local relaxation approximation of conservation laws. Consider the initial value problem (1.1) @ t u + @ x f(u) = 0 ; (1.2) u(x; 0) = u 0 (x) for (x; t) 2 IR \Theta (0; 1). Here f is a given (say C 1 ) smooth function such that f(0) = f 0 (0) = 0. Typeset by A M ST E X 2 Convergence of R...
Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation
 SIAM J. Numer. Anal
, 1997
"... Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a smallrelaxation limit governed by reduced systems of parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve and standard high resolution methods fail ..."
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Cited by 27 (11 self)
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Hyperbolic systems of conservation laws often have diffusive relaxation terms that lead to a smallrelaxation limit governed by reduced systems of parabolic or hyperbolic type. In such systems the understanding of basic wave pattern is difficult to achieve and standard high resolution methods fail to describe the right asymptotic behavior unless the small relaxation rate is numerically resolved. We develop high resolution underresolved numerical schemes that possess the discrete analogue of the continuous asymptotic limit, which thus are able to approximate the equilibrium system with high order accuracy even if the limiting equations may change type.
Diffusive Relaxation Schemes For Multiscale DiscreteVelocity Kinetic Equations
 SIAM J. NUMER. ANAL
, 1998
"... Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the NavierStokestype parabolic equations, such as the heat equation, the porous media equations, the advectiondiffusion equation, and the viscous Burgers equation. In such problems the diffusive relaxation parame ..."
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Cited by 24 (10 self)
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Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the NavierStokestype parabolic equations, such as the heat equation, the porous media equations, the advectiondiffusion equation, and the viscous Burgers equation. In such problems the diffusive relaxation parameter may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes, and it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter. Earlier approaches that work from the rarefied regimes to the Euler regimes do not directly apply to these problems since here, in addition to the stiff relaxation term, the convection term is also stiff. Our idea is to reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the convection terms into the relaxation term. This, however, introduces new difficulties du...
An AsymptoticInduced Scheme For Nonstationary Transport Equations In The Diffusive Limit
 SIAM J. Num. Anal
"... . An asymptoticinduced scheme for nonstationary transport equations with the diffusion scaling is developed. The scheme works uniformly for all ranges of mean free paths. It is based on the asymptotic analysis of the diffusion limit of the transport equation. A theoretical investigation of the beha ..."
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Cited by 24 (2 self)
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. An asymptoticinduced scheme for nonstationary transport equations with the diffusion scaling is developed. The scheme works uniformly for all ranges of mean free paths. It is based on the asymptotic analysis of the diffusion limit of the transport equation. A theoretical investigation of the behaviour of the scheme in the diffusion limit is given and an approximation property is proven. Moreover, numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically. Key words. transport equations, asymptotic analysis, diffusion limit, numerical methods for stiff equations AMS subject classifications. 82C70, 65M06, 35B25 1. Introduction. Transport equations are used to describe many physical phenomena. Some of the best known examples are neutron transport, radiative transfer equations, semiconductors or gas kinetics. The situation for small mean free paths is mathematically described by an asymptotic analysis. Depending o...
Recent Mathematical Results on Hyperbolic Relaxation Problems
, 1998
"... Contents 1 Introduction 2 2 Motivations 5 2.1 The basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The smooth case 14 3.1 Local smooth theory for quasilinear hyperbolic sys ..."
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Cited by 19 (1 self)
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Contents 1 Introduction 2 2 Motivations 5 2.1 The basic example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 The smooth case 14 3.1 Local smooth theory for quasilinear hyperbolic systems with relaxation . . . 14 3.2 Stability of global simple waves . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4 Discontinuous equilibrium solutions and weak convergence methods 26 4.1 A conservative framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 4.2 Compensated compactness results . . . . . . . . . . . . . . . . . . . . . . . . 30 4.3 Kinetic tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 5 The BV framework 39 5.1 Weakly coupled systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 5.2 Relaxation limits for the JinXin model and other discrete kinetic approximations . . . . . . . . . . . . . . . . . . . .
Diffusive Relaxation Schemes for DiscreteVelocity Kinetic Equations
 SIAM J. NUM. ANAL
"... Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the NavierStokes type parabolic equations, such as the heat equation, the porous media equations, the advectiondiffusion equation and the viscous Burgers equation. In such problems the diffusive relaxation paramet ..."
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Cited by 17 (9 self)
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Many kinetic models of the Boltzmann equation have a diffusive scaling that leads to the NavierStokes type parabolic equations, such as the heat equation, the porous media equations, the advectiondiffusion equation and the viscous Burgers equation. In such problems the diffusive relaxation parameter may differ in several orders of magnitude from the rarefied regimes to the hydrodynamic (diffusive) regimes, and it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter. Earlier approaches that work from the rarefied regimes to the Euler regimes do not directly apply to these problems since here, in addition to the stiff relaxation term, the convection term is also stiff. Our idea is to reformulate the problem in the form commonly used for the relaxation schemes to conservation laws by properly combining the stiff component of the convection terms into the relaxation term. This, however, introduces new difficulties due to t...