Results 1  10
of
43
Discrete Kinetic Schemes For Multidimensional Systems Of Conservation Laws
 SIAM J. Numer. Anal
, 2000
"... We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need ..."
Abstract

Cited by 37 (12 self)
 Add to MetaCart
We present here some numerical schemes for general multidimensional systems of conservation laws based on a class of discrete kinetic approximations, which includes the relaxation schemes by S. Jin and Z. Xin. These schemes have a simple formulation even in the multidimensional case and do not need the solution of the local Riemann problems. For these approximations we give a suitable multidimensional generalization of the Whitham's stability subcharacteristic condition. In the scalar multidimensional case we establish the rigorous convergence of the approximated solutions to the unique entropy solution of the equilibrium Cauchy problem.
A relaxation scheme for conservation laws with a discontinuous coefficient
 Math. Comp
, 2007
"... Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a wea ..."
Abstract

Cited by 32 (6 self)
 Add to MetaCart
Abstract. We study a relaxation scheme of the Jin and Xin type for conservation laws with a flux function that depends discontinuously on the spatial location through a coefficient k(x). If k ∈ BV, we show that the relaxation scheme produces a sequence of approximate solutions that converge to a weak solution. The Murat–Tartar compensated compactness method is used to establish convergence. We present numerical experiments with the relaxation scheme, and comparisons are made with a front tracking scheme based on an exact 2 × 2 Riemann solver. 1.
Numerical passage from systems of conservation laws to HamiltonJacobi equations, and relaxing schemes
 SIAM J. Numer. Anal
"... ..."
A Class of Approximate Riemann Solvers and Their Relation to Relaxation Schemes
 J. Comput. Phys
, 2001
"... We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of appro ..."
Abstract

Cited by 23 (5 self)
 Add to MetaCart
We show that a simple relaxation scheme of the type proposed by Jin and Xin [Comm. Pure Appl. Math. 48(1995) pp. 235276] can be reinterpreted as defining a particular approximate Riemann solver for the original system of m conservation laws. Based on this observation, a more general class of approximate Riemann solvers is proposed which allows as many as 2m waves in the resulting solution. These solvers are related to more general relaxation systems and connections with several other standard solvers are explored. The added flexibility of 2m waves may be advantageous in deriving new methods. Some potential applications are explored for problems with discontinuous flux functions or source terms.
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 ..."
Abstract

Cited by 19 (1 self)
 Add to MetaCart
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 . . . . . . . . . . . . . . . . . . . .
Initial Layers And Uniqueness Of Weak Entropy Solutions To Hyperbolic Conservation Laws
, 2000
"... We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weakstar in L 1 as t ! 0+ and satisfy the entropy inequality in the sense of distributions f ..."
Abstract

Cited by 19 (7 self)
 Add to MetaCart
We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weakstar in L 1 as t ! 0+ and satisfy the entropy inequality in the sense of distributions for t ? 0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss its applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation.
Pointwise Error Estimates For Relaxation Approximations to Conservation Laws
 SIAM J. Sci. Comput
, 1998
"... We obtain sharp pointwise error estimates for relaxation approximation to scalar conservation laws with piecewise smooth solutions. We first prove that the first order partial derivatives for the perturbation solutions are uniformly upper bounded (the socalled Lip stability). An onesided interpo ..."
Abstract

Cited by 16 (3 self)
 Add to MetaCart
We obtain sharp pointwise error estimates for relaxation approximation to scalar conservation laws with piecewise smooth solutions. We first prove that the first order partial derivatives for the perturbation solutions are uniformly upper bounded (the socalled Lip stability). An onesided interpolation inequality between classical L bounds enables us to convert a global L result into a (nonoptimal) local estimate. Optimal error bounds on the weighted error then follow from the maximum principle for weakly coupled hyperbolic systems. The main difficulties in obtaining the Lip stability and the optimal pointwise errors are how to construct appropriate "difference functions" so that the maximum principle can be applied. Contents 1
Convergence of singular limits for multiD semilinear hyperbolic systems to parabolic systems
 Trans. Amer. Math. Soc
, 2000
"... In this paper we investigate the diffusive zerorelaxation limit of the following multiD semilinear hyperbolic system in pseudodifferential form: W_t(x, t) + ..."
Abstract

Cited by 15 (3 self)
 Add to MetaCart
In this paper we investigate the diffusive zerorelaxation limit of the following multiD semilinear hyperbolic system in pseudodifferential form: W_t(x, t) +
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 ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
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.
Diffusion Limit of a Hyperbolic System with Relaxation
 Methods Appl. Anal
, 1998
"... In this paper we introduce a diffusive scaling to a hyperbolic system with relaxation and prove that, under such a scaling, the solution converges to that of a nonlinear convectiondiffusion equation. Using energy estimates, such a limit is justified with the initial data prescribed around a traveli ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
In this paper we introduce a diffusive scaling to a hyperbolic system with relaxation and prove that, under such a scaling, the solution converges to that of a nonlinear convectiondiffusion equation. Using energy estimates, such a limit is justified with the initial data prescribed around a traveling wave solution of the relaxation system. 1 Introduction Hyperbolic system with relaxation arises in a wide variety of physical problems, ranging from linear and nonlinear waves [22], kinetic theory [2], to multiphase and phase transition modeling. In these problems usually one can not find the exact solution. The behavior of the solution could be understood in some asymptotic regimes. In recent years much attention has been paid to the study of the zero relaxation limit. In this regime, such a complicated system may asymptotically be replaced by a much simpler hyperbolic system, the behavior of the School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. Research...