Dealing with Non-linear Terms in a Modal High-Order Discontinuous Galerkin Method

Abstract

The Discontinuous Galerkin (DG) method utilizes a mesh of elements with local functions like traditional continuous finite element methods, together with a flux approximation between elements like finite volume methods. This combination yields a high locality of the overall scheme, especially for high-order representations within elements. Two local operations need to be mainly considered. One is the application of the mass matrix and the other is the stiffness matrix. With an appropriate orthogonal basis as choice for the local functions both operations can be computed with minimal complexity. In this contribution we are concerned with a DG implementation that makes use of a Legendre polynomial basis with an application to non-linear equation systems. For non-linear systems a complication is introduced by the scheme by the necessity to compute the non-linear flux operation, which generally can not be done in the optimal modal basis. Instead, a pointwise evaluation of the non-linear operations is usually performed. Combining the fast evaluation of the integrals in the modal scheme with the pointwise evaluation of the non-linear terms requires a transformation between these two. Many methods have been developed for a fast transformation from Legendre modes to nodal values [1]. However, most of those methods for fast polynomial transformations are designed for extremely high polynomial degrees in the range of several hundreds. In three-dimensional DG simulations the polynomial degree in each dimension is more limited, and we are looking for methods that are fast but suitable for polynomials in the range up to a maximal degree of one hundred. We discuss some approaches to the fast transformation, especially the method proposed by Alpert and Rokhlin [2], and compare our implementation of this method to a straight forward \(L_2\) projection. The implementation specifically addresses also the hybrid parallelism with MPI and OpenMP for the three-dimensional DG elements.