Evaluating the stability of numerical schemes for fluid solvers in game technology

A variety of numerical techniques have been explored to solve the shallow water equations in real-time water simulations for computer graphics applications. However, determining the stability of a numerical algorithm is a complex and involved task when a coupled set of nonlinear partial differential equations need to be solved. This paper proposes a novel and simple technique to compare the relative empirical stability of finite difference (or any grid-based scheme) algorithms by solving the inviscid Burgers’ equation to analyse their respective breaking times. To exemplify the method to evaluate numerical stability, a range of finite difference schemes is considered. The technique is effective at evaluating the relative stability of the considered schemes and demonstrates that the conservative schemes have superior stability.