Solving two-phase shallow granular flow equations with a well-balanced NOC scheme on multiple GPUs

Abstract

A two-phase shallow granular flow model consists of mass and momentum equations for the solid and fluid phases, coupled together by conservative and non-conservative momentum exchange terms. Development of classic Godunov methods based on Riemann problem solutions for such a model is difficult because of complexity in building appropriate wave structures. Non-oscillatory central (NOC) differencing schemes are attractive as they do not need to solve Riemann problems. In this paper, a staggered NOC scheme is amended for numerical solution of the two-phase shallow granular flow equations due to Pitman and Le. Simple discretization schemes for the non-conservative and bed slope terms and a simple correction procedure for the updating of the depth variables are proposed to ensure the well-balanced property. The scheme is further corrected with a numerical relaxation term mimicking the interphase drag force so as to overcome the difficulty associated with complex eigenvalues in some flow conditions. The resultant NOC scheme is implemented on multiple graphics processing units (GPUs) in a server by using both OpenMP-CUDA and multistream-CUDA parallelization strategies. Numerical tests in several typical two-phase shallow granular flow problems show that the NOC scheme can model wet/dry fronts and vacuum appearance robustly, and can treat some flow conditions associated with complex eigenvalues. Comparison of parallel efficiencies shows that the multistream-CUDA strategy can be slightly faster or slower than the OpenMP-CUDA strategy depending on the grid sizes.