Discontinuous Galerkin method on three-dimensional tetrahedral grids: Using the operator programming method

Abstract

In the numerical simulation of gas-dynamic flows in domains with a complex geometry, it is necessary to use detailed unstructured grids and highly accurate numerical methods. The Galerkin method with discontinuous base functions (or the discontinuous Galerkin method) works well in dealing with such problems. This technique has several advantages inherent both in finite-element and in finite-difference approximations. At the same time, the discontinuous Galerkin method is computationally complex; therefore, the question arises about the most efficient use of the full potential of computers. In order to speed up the computations, we applied the operator programming method to develop the computational module. It allows presenting mathematical formulas in programs in compact form and helps to port programs to parallel architectures such as NVidia CUDA and Intel Xeon Phi. Earlier the operator programming method was implemented for regular three-dimensional Cartesian grids and three-dimensional locally adaptive grids. In this work, this method is applied to threedimensional tetrahedral grids. This example demonstrates that the method in question can be efficiently implemented on arbitrary three-dimensional grids. Besides, we demonstrate the use of the template metaprogramming methods of the C++ programming language in order to speed up computations.