Second Order Accuracy of Brenier’s Time-Discrete Method for Nonlinear Systems of Conservation Laws

1988 ◽  
Vol 25 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Randall J. LeVeque
Keyword(s):  
Nonlinear Systems ◽  
Conservation Laws ◽  
Second Order ◽  
Order Accuracy ◽  
Discrete Method ◽  
Systems Of Conservation Laws

A weighted average flux method for hyperbolic conservation laws

1989 ◽  
Vol 423 (1865) ◽  
pp. 401-418 ◽  
Keyword(s):  
Nonlinear Systems ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Model Equation ◽  
Hyperbolic Conservation Laws ◽  
Weighted Average ◽  
Second Order ◽  
Order Accuracy ◽  
Hyperbolic Conservation ◽  
Average Flux

A numerical technique, called a ‘weighted average flux’ (WAF) method, for the solution of initial-value problems for hyperbolic conservation laws is presented. The intercell fluxes are defined by a weighted average through the complete structure of the solution of the relevant Riemann problem. The aim in this definition is the achievement of higher accuracy without the need for solving ‘generalized’ Riemann problems or adding an anti-diffusive term to a given first-order upwind method. Second-order accuracy is proved for a model equation in one space dimension; for nonlinear systems second-order accuracy is supported by numerical evidence. An oscillation-free formulation of the method is easily constructed for a model equation. Applications of the modified technique to scalar equations and nonlinear systems gives results of a quality comparable with those obtained by existing good high resolution methods. An advantage of the present method is its simplicity. It also has the potential for efficiency, because it is well suited to the use of approximations in the solution of the associated Riemann problem. Application of WAF to multidimensional problems is illustrated by the treatment using dimensional splitting of a simple model problem in two dimensions.

Generalised characteristics in hyperbolic systems of conservation laws with special coupling

1990 ◽  
Vol 116 (3-4) ◽  
pp. 245-278 ◽  
Author(s):  
C. M. Dafermos ◽  
X. Geng
Keyword(s):  
Nonlinear Systems ◽  
Conservation Laws ◽  
Rarefaction Wave ◽  
Hyperbolic Systems ◽  
A Priori ◽  
Systems Of Conservation Laws ◽  
Straight Lines

SynopsisUsing the theory of generalised characteristics, we study the structure of BV solutions of genuinely nonlinear systems of two conservation laws whose shock and rarefaction wave curves of the first family are straight lines. We also establish a priori estimates on the variation of the solution similar to those obtained earlier by Glimm and Lax.

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