On the standard Galerkin method with explicit RK4 time stepping for the Shallow Water equations
Provides rigorous error analysis for a common numerical method applied to shallow water equations, but the results are incremental as they extend existing theory to a specific combination of discretization schemes.
The authors prove L2 error estimates for the standard Galerkin method with explicit RK4 time stepping for 1D shallow water equations, achieving fourth-order temporal accuracy and suboptimal spatial order on quasiuniform meshes. Numerical experiments confirm the convergence rates.
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.