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Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. (English) Zbl 0860.65089
A hyperbolic system of conservation laws with relaxation term $$q(u)$$ has the form $$\partial_t u+\partial_xF(u)=q(u)$$. The article concerns the numerical methods for such systems. The objectives are twofold. Firstly, it is shown that when stiff relaxation terms are present, most shock-capturing schemes (like the upwind scheme, the van Leer scheme, the piecewise parabolic method, piecewise steady approximation, etc.) fail to capture the proper parabolic long-time behavior for coarse grids. This phenomenon is demonstrated separately for the linear telegraph equations, a convection-diffusion equation and some more general $$2\times 2$$ systems.
Secondly, the authors develop for such systems semidiscrete numerical schemes (a modification of higher order Godunov ones) that give the correct long-time behavior. These methods are based on the asymptotic analysis of the behavior of the solution that shows a balance between the relaxation terms and spatial derivative terms. Several experiments with numerical results illustrate the analysis.

##### MSC:
 65M20 Method of lines for initial value and initial-boundary value problems involving PDEs 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs 35L65 Hyperbolic conservation laws
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