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 [8] is implemented for solving 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 [22], 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.

Application of a Modified Parareal Method for Speeding Up the Numerical Resolution of the 2D Shallow Water Equations

“…( 1a) -(1c)) are discretized using a finite volume (FV) scheme and an explicit Euler temporal evolution. We refer to [14] for a detailed description of the model reduction of (1a)-(1c) using the procedures described in Section 3. All the parareal simulations are performed with a maximum number of iterations 𝑁 itermax = 5 (except when unstable behaviors lead to negative water depth and stop the numerical resolution).…”