Alternative Solvers for the Derivative Riemann Problem for Hyperbolic Balance Laws

We study methods for solving the Cauchy problem for systems of non-linear hyperbolic balance laws with initial condition consisting of two smooth vectors, with a discontinuity at the origin. We call this initial-value problem the Derivative Riemann Problem, or DRP. Two new methods of solution are presented. The first one results from a re-interpretation of the high-order numerical methods proposed by Harten et al. [10] and the second method is a modification of the DRP solver in [35]. A systematic assessment of all available DRP solvers is carried out and their relative merits are discussed. Finally, we also implement the DRP solvers, locally, in the context of high-order finite volume numerical methods of the ADER type, on unstructured meshes. Schemes of up to fifth order of accuracy in space and time for the two-dimensional compressible Euler equations are constructed. Empirically obtained convergence rates are studied systematically and, for the tests considered, these correspond to the theoretically expected orders of accuracy.