An hp ‐adaptive discontinuous Galerkin method for shallow water flows

An adaptive spectral/ hp discontinuous Galerkin method for the two‐dimensional shallow water equations is presented. The model uses an orthogonal modal basis of arbitrary polynomial order p defined on unstructured, possibly non‐conforming, triangular elements for the spatial discretization. Based on a simple error indicator constructed by the solutions of approximation order p and p −1, we allow both for the mesh size, h , and polynomial approximation order to dynamically change during the simulation. For the h ‐type refinement, the parent element is subdivided into four similar sibling elements. The time‐stepping is performed using a third‐order Runge–Kutta scheme. The performance of the hp ‐adaptivity is illustrated for several test cases. It is found that for the case of smooth flows, p ‐adaptivity is more efficient than h ‐adaptivity with respect to degrees of freedom and computational time. Copyright © 2010 John Wiley & Sons, Ltd.