Riemann solutions for spacetime discontinuous Galerkin methods

Spacetime discontinuous Galerkin finite element methods (cf. Abedi et al. (2013), Abedi et al. (2010), Miller and Haber (2008)) rely on ‘target fluxes’ on element boundaries that are computed via local one-dimensional Riemann solutions in the direction normal to the element face. In this work, we provide details of converting a space–time flux expressed in differential forms into a standard one-dimensional Riemann problem on the element interface. We then demonstrate a generalized solution procedure for linearized hyperbolic systems based on diagonalization of the governing system of partial differential equations. The generalized procedure is particularly useful for the implementation aspects of coupled multi-physics applications. We show that source terms do not influence the Riemann solution in the spacetime setting. We provide details for implementation of coordinate transformations and Riemann solutions. Exact Riemann solutions for some linearized systems of equations are provided as examples, including an exact, semi-analytic Riemann solution for generalized thermoelasticity with one relaxation time.