A high-order triangular discontinuous Galerkin oceanic shallow water model
Abstract
A high-order triangular discontinuous Galerkin (DG) method is applied to the two-dimensional oceanic
shallow water equations. The DG method can be characterized as the fusion of finite elements with finite
volumes. This DG formulation uses high-order Lagrange polynomials on the triangle using nodal sets
up to 15th order. Both the area and boundary integrals are evaluated using order 2N Gauss cubature
rules. The use of exact integration for the area integrals leads naturally to a full mass matrix; however,
by using straight-edged triangles we eliminate the mass matrix completely from the discrete equations.
Besides obviating the need for a mass matrix, triangular elements offer other obvious advantages in the
construction of oceanic shallow water models, specifically the ability to use unstructured grids in order to
better represent the continental coastlines for use in tsunami modeling. In this paper, we focus primarily on
testing the discrete spatial operators by using six test cases—three of which have analytic solutions. The
three tests having analytic solutions show that the high-order triangular DG method exhibits exponential
convergence. Furthermore, comparisons with a spectral element model show that the DG model is superior
for all polynomial orders and test cases considered.
Description
The article of record as published may be found at http://dx.doi.org/10.1002/fld.1562