Numerical schemes for hyperbolic conservation laws with stiff relaxation terms.

*(English)*Zbl 0860.65089A 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.

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.

Reviewer: S.Migorski (Krakow)

##### 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 |