The Riemann problem in computational science

A review of the Riemann problem and its impact in computational science is given, starting from basic definitions and simple examples. I then concentrate on approximate Riemann solvers for use in numerical methods, such as finite volume and discontinuous Galerkin finite element methods. A brief overview of the most well known approximate methods is given. There follows a review of some specific approximate Riemann solvers in more detail; I have chosen examples that are connected to the author’s own research and have included some recent developments. The paper ends with a discussion on generalisations of the Riemann problem, to account for source terms and piece-wise smooth initial data, and the construction of very-high order ADER methods based on the generalised Riemann problem