Modified parareal method for solving the two-dimensional nonlinear shallow water equations using finite volumes

Abstract

In this work, the POD-DEIM-based parareal method introduced in [6] is implemented for the resolution of the two-dimensional nonlinear shallow water equations using a finite volume scheme. This method is a variant of the traditional parareal method, first introduced by [19], that improves the stability and convergence for nonlinear hyperbolic problems, and uses reduced-order models constructed via the Proper Orthogonal Decomposition-Discrete Empirical Interpolation Method (POD-DEIM) applied to snapshots of the solution of the parareal iterations. We propose a modification of this parareal method for further stability and convergence improvements. It consists in enriching the snapshots set for the POD-DEIM procedure with extra snapshots whose computation does not require any additional computational cost. The performances of the classical parareal method, the POD-DEIM-based parareal method and our proposed modification are compared using numerical tests with increasing complexity. Our modified method shows a more stable behaviour and converges in fewer iterations than the other two methods.