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

Abstract

In this work, we implement some variations of the parareal method for speeding up the numerical resolution of the two-dimensional nonlinear shallow water equations (SWE). This method aims to reduce the computational time required for a fine and expensive model, by using alongside a less accurate, but much cheaper, coarser one, which allows to parallelize in time the fine simulation. We consider here a variant of the method using reduced-order models and suitable for treating nonlinear hyperbolic problems, being able to reduce stability and convergence issues of the parareal algorithm in its original formulation. We also propose a modification of the ROM-based parareal method consisting in the enrichment of the input data for the model reduction with extra information not requiring any additional computational cost to be obtained. Numerical simulations of the 2D nonlinear SWE with increasing complexity are presented for comparing the configurations of the model reduction techniques and the performance of the parareal variants. Our proposed method presents a more stable behavior and a faster convergence towards the fine, referential solution, providing good approximations with a reduced computational cost. Therefore, it is a promising tool for accelerating the numerical simulation of problems in hydrodynamics.