Multi-dimensional Riemann problems for linear hyperbolic systems

RIEMANN PROBLEMS FOR NONSTRICTLY HYPERBOLIC 2x2 SYSTEMS OF CONSERVATION LAWS

The Riemann problem is solved for 2 x 2 systems of hyperbolic conservation laws having quadratic flux functions. Equations with quadratic flux functions arise from neglecting higher order nonlinear terms in hyperbolic systems that fail to be strictly hyperbolic everywhere. Such equations divide into four classes, three of which are considered in this paper. The solution of the Riemann problem i...

The hyperbolic/elliptic transition in the multi-dimensional Riemann Problem

For a continuous self-similar solution to a system of conservation laws, genuine nonlinearity yields Lipschitz continuity at points where the type of the governing system changes. This is a well-known fact in one space dimension, where a constant state ū bifurcates towards a rarefaction wave at a point x/t that equals an eigenvalue λj(ū). We extend this observation to several space dimensions. ...

Boundary Conditions for Multi-dimensional Hyperbolic Relaxation Problems

Exponential Dichotomy for Asymptotically Hyperbolic Two-Dimensional Linear Systems

We consider the problem of determing the existence of exponential dichotomy for a class of linear nonautonomous ODEs. An approach is introduced that combines numerical techniques with rigorous perturbation theory. It is applicable to a given problem within the class we consider and for practical purposes we develop a continuation technique. Numerical results illustrate the utility of the approach.

Self-similar solutions‎ ‎of the Riemann problem for two-dimensional systems of conservation‎ ‎laws

In this paper, a new approach is applied to study the self-similar solutions of 2×2 systems of nonlinear hyperbolic conservation laws. A notion of characteristic directions is introduced and then used to construct local and smooth solutions of the associated Riemann problem