scholarly journals Four-Quadrant Riemann Problem for a 2×2 System Involving Delta Shock

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 138
Author(s):  
Jinah Hwang ◽  
Suyeon Shin ◽  
Myoungin Shin ◽  
Woonjae Hwang
Keyword(s):  
Numerical Solution ◽  
Riemann Problem ◽  
Hyperbolic Conservation Laws ◽  
Analytic Solution ◽  
Hyperbolic Conservation ◽  
Delta Shock ◽  
Initial Discontinuity ◽  
Planar Wave ◽  
Four Quadrant ◽  
Elementary Waves

In this paper, a four-quadrant Riemann problem for a 2×2 system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave at each initial discontinuity. We include the existence of two elementary waves at each initial discontinuity and classify 14 topologically distinct solutions. For each case, we construct an analytic solution and compute the numerical solution.

scholarly journals Four-Quadrant Riemann Problem for a 2×2 System II

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 592
Author(s):  
Jinah Hwang ◽  
Suyeon Shin ◽  
Myoungin Shin ◽  
Woonjae Hwang
Keyword(s):  
Initial Data ◽  
Riemann Problem ◽  
Numerical Solutions ◽  
Analytic Solutions ◽  
Wave Interactions ◽  
Delta Shock ◽  
Elementary Wave ◽  
Initial Discontinuity ◽  
The Rich ◽  
Four Quadrant

In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions.

Riemann problem for a class of 2 × 2 hyperbolic conservation laws

1996 ◽  
Vol 126 (4) ◽  
pp. 719-724 ◽  
Author(s):  
Changjiang Zhu
Keyword(s):  
Global Existence ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Hyperbolic Conservation Laws ◽  
Hyperbolic Conservation

In this paper we prove the global existence of the solutions of the Riemann problem for a class of 2 × 2 hyperbolic conservation laws, which is neither necessarily strictly hyperbolic nor necessarily genuinely nonlinear.

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.

Sign in / Sign up

Export Citation Format

Share Document