High-Order Schemes for the Shallow Water Equations on GPUs

Abstract

In this thesis, well-balanced, central-upwind high-resolution methods of high order are developed for the two-dimensional shallow water equations, on the graphics processing unit (GPU). The methods are based on a fifth-order Weighted Essentially Non-Oscillating (WENO) reconstruction technique and a fourth-order Gaussian quadrature for the one-sided interface fluxes. Two schemes are implemented, one with bilinear interpolation of the bottom topography and a second-order quadrature for the bed slope source term, and one with a fourth-order source term quadrature and a fifth-order hydrostatic WENO reconstruction of the water height. The high-order schemes are compared to a second-order scheme by Kurganov and Petrova, which recently has been implemented on the GPU by Brodtkorb. The schemes are shown to be well-balanced and are capable of outperforming the second-order scheme on smooth problems provided that dry states and discontinuities do not occur. The performance gain is larger after a low number of time steps, where speed-ups by a factor between 3 and 8 are documented. As the system evolves, the accuracy of the high-order methods drops, and the performance gain is reduced to a factor around 1.5. The schemes do, however, not support outflow and inflow boundary conditions, and are not yet tested on real-world problems. The high-order scheme using the hydrostatic reconstruction and a fourth-order source term quadrature is implemented in such a way that an extension to quadratures and reconstructions of even higher order, should be fairly straight forward.