An augmented HLLEM ADER numerical model parallel on GPU for the porous Shallow Water Equations

This paper presents a novel explicit finite volume scheme for the solution of the two-dimensional Shallow Water Equations (SWEs) with porosity based on the use of an augmented Riemann problem. Bottom and porous source terms are treated as non-conservative fluxes by including two additional equations to the porous SWEs. The HLLEM approximate Riemann solver is chosen to compute the fluxes of the non-conservative system of equations in order to ensure robustness, computational efficiency and entropy enforcement also in the presence of significant discontinuities in the porosity field. Moreover, the proposed solver is extended to second order in time and space adopting the ADER time-integration scheme, together with a Total Variation Diminishing (TVD) spatial reconstruction. The model is implemented in CUDA (Compute Unified Device Architecture) language to exploit the computational power of modern Graphic Processing Units. The theoretical and numerical verification of the C-property is achieved also when discontinuities characterize the porosity and bottom fields. The scheme accurately reproduces Riemann problems generated by the presence of porous discontinuities. Numerical tests also confirm that the proposed algorithm retains roughly the same numerical efficiency of an alternative formulation based on a non-augmented Riemann solver, while providing more accurate results.