Efficient GPU Implementation of Multidimensional Incomplete Riemann Solvers for Hyperbolic Nonconservative Systems: Applications to Shallow Water Systems with Topography and Dry Areas

This paper deals with a class of efficient, genuinely two-dimensional Riemann solvers for hyperbolic nonconservative systems. A particularity of these solvers is that only a bound on the maximal propagation speeds in the coordinate directions is needed. The amount of numerical diffusion is easily controlled by an appropriate choice of the numerical viscosity matrix. Special attention has been given to applications to the one-layer and two-layer shallow water systems, including topography and dry areas. The purpose of the paper is two-fold. On the one hand, we describe an extension of the numerical schemes previously introduced in Schneider et al. (J Comput Phys 444:110547, 2021) to correctly handle the existence of wet-dry transition in the computational domain, maintaining at the same time the well-balancing properties of the schemes. On the other hand, we propose an efficient implementation of the schemes in Graphical Processing Units (GPUs), making the schemes highly competitive when confronted with challenging problems. The performances of the schemes have been tested with some numerical experiments, showing that they are, in general, more efficient than their \(\hbox {1d}\times \hbox {1d}\) versions.