An Adaptive Stable Space-Time FE Method for the Shallow Water Equations

TL;DR

This paper introduces an unconditionally stable finite element method for the shallow water equations that employs adaptive mesh refinement and space-time slicing to improve accuracy and computational efficiency.

Contribution

It develops an adaptive, unconditionally stable FE method using the AVS-FE approach for the SWE, with error estimation and space-time adaptivity for enhanced simulation accuracy.

Findings

The AVS-FE method achieves asymptotic convergence for the SWE.

Space-time slices outperform full domain simulations over long times.

Adaptive mesh refinement effectively improves solution accuracy.

Abstract

We consider the finite element (FE) approximation of the shallow water equations (SWE) by considering discretizations in which both space and time are established using an unconditionally stable FE method. Particularly, we consider the automatic variationally stable FE (AVS-FE) method, a type of discontinuous Petrov-Galerkin (DPG) method. The philosophy of the DPG method allows us to break the test space and achieve unconditionally stable FE approximations as well as accurate a posteriori error estimators upon solution of a saddle point system of equations. The resulting error indicators allow us to employ mesh adaptive strategies and perform space-time mesh refinements, i.e., local time stepping. We derive a priori error estimates for the AVS-FE method and linearized SWE and perform numerical verifications to confirm corresponding asymptotic convergence behavior. In an effort to keep the computational cost low, we consider an alternative space-time approach in which the space-time domain is partitioned into finite sized space-time slices. Hence, we can perform adaptivity on each individual slice to preset error tolerances as needed for a particular application. Numerical verifications comparing the two alternatives indicate the space-time slices are superior for simulations over long times, whereas the solutions are indistinguishable for short times. Multiple numerical verifications show the adaptive mesh refinement capabilities of the AVS-FE method, as well the application of the method to commonly applied benchmarks for the SWE.