High-Order Discontinuous Galerkin Methods using a Spectral Multigrid Approach

Abstract

The goal of this paper is to investigate and develop a fast and robust algorithm for the solution of high-order accurate Discontinuous Galerkin discretizations of non-linear systems of conservation laws on unstructured grids. Herein we present the development of a spectral hp-multigrid method, where the coarse “grid” levels are constructed by reducing the order (p) of approximation of the discretization using hierarchical basis functions ( p-multigrid), together with the traditional (h-multigrid) approach of constructing coarser grids with fewer elements. On each level we employ variants of the element-Jacobi scheme, where the Jacobian entries associated with each element are inverted directly and all other entries are treated explicitly. The methodology is developed for the non-linear Euler equations, using both non-linear (FAS) and linear (CGC) multigrid schemes, and results are presented for the channel flow over a bump and a four element airfoil. Current results demonstrate convergence rates which are independent of the order of accuracy (p) of the discretization, with slight dependence on the level of mesh resolution ( h).