An hp‐adaptive discontinuous Galerkin method for shallow water flows

Abstract

We have extended Cosmos++, a multi-dimensional unstructured adaptive mesh code for solving the covariant Newtonian and general relativistic radiation magnetohydrodynamic (MHD) equations, to accommodate both discrete finite volume and arbitrarily high order finite element structures. The new finite element implementation, called CosmosDG, is based on a discontinuous Galerkin (DG) formulation, using both entropy-based artificial viscosity and slope limiting procedures for regular-ization of shocks. High order multi-stage forward Euler and strong stability preserving Runge-Kutta time integration options complement high order spatial discretization. We have also added flexibility in the code infrastructure allowing for both adaptive mesh and adaptive basis order refinement to be performed separately or simultaneously in a local (cell-by-cell) manner. We discuss in this report the DG formulation and present tests demonstrating the robustness, accuracy, and convergence of our numerical methods applied to special and general relativistic MHD, though we note an equivalent capability currently also exists in CosmosDG for Newtonian systems.

Analyzing the Adaptive Mesh Refinement (AMR) Characteristics of a High-Order 2D Cubed-Sphere Shallow-Water Model

“…Use of p refinement with discontinuous Galerkin (DG) methods for the shallowwater equations were described by Kubatko et al (2009) and Tumolo and Bonaventura (2015). A hybrid refinement method that combines the h-and p-refinements methods was demonstrated by Eskilsson (2011), and Blaise and St-Cyr (2012) used an hp-adaptive DG method to model the shallow-water equations on the sphere for global tsunami simulations. Recently, Aechtner et al (2015) implemented a new adaptive wavelet approach for local dynamic refinement with the 2D shallow-water equations on the sphere.…”

Analysis of adaptive mesh refinement for IMEX discontinuous Galerkin solutions of the compressible Euler equations with application to atmospheric simulations

“…[34] shows the application of such an algorithm for the DG method using triangular, non-conforming elements applied to shallow water equations. Another example for this set of equations is presented in [35], where an hp-adaptive DG method on quadrilateral, non-conforming elements for global tsunami simulations is considered.…”

A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations

“…[46][47][48][49]. The technique we employ is extremely simple and relies on the use of orthogonal hierarchical tensor-product basis functions.…”