Strong and Weak Lagrange-Galerkin Spectral Element Methods for the Shallow Water Equations

The Lagrange-Galerkin spectral element method for the two-dimensional shallow water equations is presented. The equations are written in conservation form and the domains are discretized using quadrilateral elements. Lagrangian methods integrate the governing equations along the characteristic curves, thus being well suited for resolving the nonlinearities introduced by the advection operator of the fluid dynamics equations. Two types of Lagrange-Galerkin methods are presented the strong and weak formulations. The strong form relies mainly on interpolation to achieve high accuracy while the weak form relies primarily on integration. Lagrange-Galerkin schemes offer an increased efficiency by virtue of their less stringent CFL condition. The use of quadrilateral elements permits the construction of spectral-type finite-element methods that exhibit exponential convergence as in the conventional spectral method yet they are constructed locally as in the finite-element method this is the spectral element method. In this paper, we show how to fuse the Lagrange-Galerkin methods with the spectral element method and present results for two standard test cases in order to compare and contrast these two hybrid schemes.