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

D. c. Antonopoulos; V. a. Dougalis; G. Kounadis

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arXiv · 1810.11008

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

D. c. Antonopoulos·V. a. Dougalis·G. Kounadis

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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.

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D. c. Antonopoulos, V. a. Dougalis, G. Kounadis. 2018-10-25. On the standard Galerkin method with explicit RK4 time stepping for the Shallow Water equations. https://arxiv.org/abs/1810.11008

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