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