The overarching goal of numerical simulation is to offer the most accurate solution in the shortest amount of time, and typically also using the least amount of computational resources. Whilst the accuracy and convergence of numerical schemes often is well studied, there is little literature that on how such numerical schemes compare in practice on physical hardware.
The aim of this paper is to compare the accuracy and computational efficiency on GPUs of finite volume schemes for the shallow water equations. At the end of the day, it is time-to-result with a certain accuracy that counts, and it is not given which numerical scheme yields the best results for this metric.
We compare the efficiency and accuracy of several different finite volume schemes on GPUs for one-dimensional and two-dimensional problems, with both shock and smooth solutions. The selected schemes include Lax–Friedrichs, FORCE, Harten–Lax–van Leer (HLL), Kurganov–Petrova, and Weighted Average Flux.
We compare the computing time for each scheme with a predetermined accuracy requirement, and vice versa how accurate a numerical scheme is given a set computing time.