Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

Wang, Weicheng; Xin, Zhouping

doi:10.1002/(sici)1097-0312(199805)51:5<505::aid-cpa3>3.0.co;2-c

Cited by 56 publications

(36 citation statements)

References 9 publications

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“…W.C. Wang and Z. Xin [46] show the existence and uniqueness of the solution of (12)(13). Convergence holds under strong restrictions: the initial boundary data must be a small perturbation of a constant state u * , which is supposed to be non transonic (i.e.…”

Section: Entropy Flux Pairmentioning

confidence: 99%

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Kinetic approximation of a boundary value problem for conservation laws

Aregba-Driollet

Milisic

2004

Numer. Math.

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We design numerical schemes for systems of conservation laws with boundary conditions. These schemes are based on relaxation approximations taking the form of discrete BGK models with kinetic boundary conditions. The resulting schemes are Riemann solver free and easily extendable to higher order in time or in space. For scalar equations convergence is proved. We show numerical examples, including solutions of Euler equations.

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Section: Entropy Flux Pairmentioning

confidence: 99%

“…From the work of Z. Xin and W.C. Wang [46], it results that stability and uniqueness were obtained under the strong constraint that the data must be a small perturbation of a constant nontransonic state.…”

Section: The Scalar Casementioning

confidence: 99%

Kinetic approximation of a boundary value problem for conservation laws

Aregba-Driollet

Milisic

2004

Numer. Math.

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“…Based on a general framework developed in [23,25], the first-order rate of convergence for (1.1) is established in the case when its equilibrium solutions are piecewise smooth [24], which is an improvement on the O( √ ) error bounds [7,8]. The boundary layer effect in the small relaxation limit to the equilibrium scalar conservation laws was investigated in [27]. The existence and uniqueness for the initial-boundary value problems are established.…”

Section: Introductionmentioning

confidence: 99%

Pointwise Error Estimates for Relaxation Approximations to Conservation Laws

Tadmor

Tang²

2000

SIAM J. Math. Anal.

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Abstract. 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 so-called Lip + stability). A one-sided interpolation inequality between classical L 1 error estimates and Lip + stability bounds enables us to convert a global L 1 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.

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“…The existence and stability of shock profiles (traveling waves) for a general nonlinear n × n system with relaxation were studied in [52] and [31], respectively. For the initial boundary value problem, one can refer to [34], [41], [46], [47] and [48]. A quasilinear model of gas dynamics equations was investigated in [53].…”

Section: Haitao Fan and Tao Luomentioning

confidence: 99%

Convergence to equilibrium rarefaction waves for discontinuous solutions of shallow water wave equations with relaxation

Fan

Luo

2005

Quart. Appl. Math.

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Abstract. The purpose of this paper is to study the discontinuous solutions to a shallow water wave equation with relaxation. The typical initial value problem of discontinuous solutions is the Riemann problem. Unlike the homogeneous hyperbolic conservation laws, due to the inhomogeneity of the system studied here, the solutions of the Riemann problem do not have a self-similar structure anymore. This problem can be formulated as a free boundary problem. We show that the Riemann solutions still have a piecewise smooth structure globally and converge to the rarefaction waves of the equilibrium equation as time tends to infinity.

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Asymptotic limit of initial boundary value problems for conservation laws with relaxational extensions

Cited by 56 publications

References 9 publications

Kinetic approximation of a boundary value problem for conservation laws

Kinetic approximation of a boundary value problem for conservation laws

Pointwise Error Estimates for Relaxation Approximations to Conservation Laws

Convergence to equilibrium rarefaction waves for discontinuous solutions of shallow water wave equations with relaxation

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