scholarly journals Approximate solutions of generalized Riemann problems for nonlinear systems of hyperbolic conservation laws

2015 ◽  
Vol 85 (297) ◽  
pp. 35-62 ◽  
Author(s):  
Claus R. Goetz ◽  
Armin Iske
Keyword(s):  
Nonlinear Systems ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Approximate Solutions ◽  
Riemann Problems ◽  
Hyperbolic Conservation ◽  
Generalized Riemann Problems

LARGE TIME BEHAVIOR OF NUMERICAL SOLUTIONS OF SCALAR CONSERVATION LAWS

2006 ◽  
Vol 03 (04) ◽  
pp. 631-648
Author(s):  
FRÉDÉRIC LAGOUTIÈRE
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Large Time ◽  
Numerical Solutions ◽  
Approximate Solutions ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Hyperbolic Conservation ◽  
Periodic Initial Data ◽  
Scalar Conservation

We study the large time behavior of entropic approximate solutions to one-dimensional, hyperbolic conservation laws with periodic initial data. Under mild assumptions on the numerical scheme, we prove the asymptotic convergence of the discrete solutions to a time- and space-periodic solution.

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.

AN ALTERNATING EVOLUTION APPROXIMATION TO SYSTEMS OF HYPERBOLIC CONSERVATION LAWS

2008 ◽  
Vol 05 (02) ◽  
pp. 421-447 ◽  
Author(s):  
HAILIANG LIU
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Entropy Solution ◽  
Approximate Solutions ◽  
Spatial Dimension ◽  
Approximation Formula ◽  
Method Error ◽  
Hyperbolic Conservation ◽  
Spatial Redistribution ◽  
Systems Of Conservation Laws

In this paper, we present an alternating evolution (AE) approximation [Formula: see text] to systems of hyperbolic conservation laws [Formula: see text] in arbitrary spatial dimension. We prove the convergence of the approximate solutions towards an entropy solution of scalar multi-D conservation laws, and the L1 contraction property for the approximate solution is established as well. It is also shown that such an approximation is extremely accurate in the sense that if initial data is prepared such that u0 = v0 = U0, then no method error is induced as time evolves, and the exact entropy solution is precisely captured. Furthermore, in the approximation system time evolution of one variable is associated with spatial redistribution in another variable. These features render such an approximation ideal to be used for construction of high resolution numerical schemes to solve hyperbolic conservation laws. The usual obstacles caused by jumps crossing computational cell interfaces are not felt when both u and v are sampled alternatively, and reconstructed independently. Herewith we discuss the designing principle for constructing AE schemes, with illustration of two preliminary schemes for systems of conservation laws in one dimension. Both l∞ monotonicity and the TVD (Total Variational Diminishing) property are established for these schemes when applied to the scalar laws.

scholarly journals High order explicit local time stepping methods for hyperbolic conservation laws

2020 ◽  
Vol 89 (324) ◽  
pp. 1807-1842
Author(s):  
Thi-Thao-Phuong Hoang ◽  
Lili Ju ◽  
Wei Leng ◽  
Zhu Wang
Keyword(s):  
Conservation Laws ◽  
Local Time ◽  
Hyperbolic Conservation Laws ◽  
High Order ◽  
Hyperbolic Conservation ◽  
Time Stepping ◽  
Local Time Stepping
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