Numerical Capture of Shock Solutions of Nonconservative Hyperbolic Systems via Kinetic Functions

This paper reviews recent contributions to the numerical approximation of solutions of nonconservative hyperbolic systems with singular viscous perturbations. Various PDE models for complex compressible materials enter the proposed framework. Due to lack of a conservative form in the limit systems, associated weak solutions are known to heavily depend on the underlying viscous regularization. This small scales sensitiveness drives the classical approximate Riemann solvers to grossly fail in the capture of shock solutions. Here, small scales sensitiveness is encoded thanks to the notion of kinetic functions so as to consider a set of generalized jump conditions. To enforce for validity these jump conditions at the discrete level, we describe a systematic and effective correction procedure. Numerical experiments assess the relevance of the proposed method.