Stability analysis and convergence rate of a two-step predictor-corrector approach for shallow water equations with source terms

Abstract

This paper deals with a two-step explicit predictor-corrector approach so-called the two-step MacCormack formulation, for solving the one-dimensional nonlinear shallow water equations with source terms. The proposed two-step numerical scheme uses the fractional steps procedure to treat the friction slope and to upwind the convection term in order to control the numerical oscillations and stability. The developed scheme uses both forward and backward difference formulations in the predictor and corrector steps, respectively. The linear stability of the constructed technique is deeply analyzed using the Von Neumann stability approach whereas the convergence rate of the proposed method is numerically obtained in the $ L^{2} $-norm. A wide set of numerical examples confirm the theoretical results.