Towards a compact high-order method for non-linear hyperbolic systems, II. The Hermite-HLLC scheme

In a finite-volume framework, we develop an approximate HLL Riemann solver specific to weakly hyperbolic systems. Those systems are obtained by considering not only the variable but also its first spatial derivative, as unknowns. To this aim, we rely upon the theory of “δ-shock waves”, newly developed in the scalar case.First, we demonstrate that the extended version of the HLLE scheme to weakly hyperbolic systems is compatible with the existence of Dirac measures in the solution. Then, we develop a specific Hermite Least-Square (HLSM) interpolation that enables to generate a high-order and compact scheme, without creating spurious oscillations in the reconstruction of the variable or its first derivative. Extensive numerical experiments make it possible to validate the method and to check convergence to entropy solutions.Relying upon those results, we construct a new HLL Riemann solver, suited for the extended one-dimensional Euler equations. For this purpose, we introduce the contribution of a contact discontinuity inside the definition of the solver. By using a formal analogy with the scalar study, we demonstrate that this solver tolerates the existence of “δ-shock waves” in the solution. Numerical experiments that follow help to validate some of the assumptions made to generate this scheme.