Derivative Riemann solvers for systems of conservation laws and ADER methods

In this paper, we first briefly review the semi-analytical method [E.F. Toro, V.A. Titarev, Solution of the generalized Riemann problem for advection–reaction equations, Proc. Roy. Soc. London 458 (2018) (2002) 271–281] for solving the derivative Riemann problem for systems of hyperbolic conservation laws with source terms. Next, we generalize it to hyperbolic systems for which the Riemann problem solution is not available. As an application example we implement the new derivative Riemann solver in the high-order finite-volume ADER advection schemes. We provide numerical examples for the compressible Euler equations in two space dimensions which illustrate robustness and high accuracy of the resulting schemes.