On the standard Galerkin method with explicit RK4 time stepping for the Shallow Water equations

TL;DR

This paper analyzes the accuracy of combining the standard Galerkin finite element method with explicit RK4 time stepping for the 1D shallow water equations, providing error estimates and computational validation.

Contribution

It offers the first rigorous L2 error estimates for this combined method, demonstrating fourth-order temporal accuracy and suboptimal spatial accuracy under certain conditions.

Findings

Fourth-order temporal accuracy confirmed

Suboptimal spatial convergence observed

Computational validation supports theoretical error estimates

Abstract

We consider a simple initial-boundary-value problem for the shallow water equations in one space dimension. We discretize the problem in space by the standard Galerkin finite element method on a quasiuniform mesh and in time by the classical 4-stage, 4th order, explicit Runge-Kutta scheme. Assuming smoothness of solutions, a Courant number restriction, and certain hypotheses on the finite element spaces, we prove L2 error estimates that are of fourth-order accuracy in the temporal variable and of the usual, due to the nonuniform mesh, suboptimal order in space. We also make a computational study of the numerical spatial and temporal orders of convergence, and of the validity of a hypothesis made on the finite element spaces.