Discussion of Python Implementation Techniques for Discontinuous Galerkin Methods

Abstract: This paper discusses implementation techniques for integration methods within the Discontinuous Galerkin Methods. These methods are used to approximate solutions for differential equations. To do so, one must compute a polynomial, which is an approximation of the used function. To compute this approximation, orthogonal polynomials, chosen to be the Legendre polynomials are used, as well as integration. This is first demonstrated for one dimension. For two or three dimensions, full tensor product or sum-factorization is used. These implementations give the same approximation of the polynomial, but sum-factorization uses fewer computations than full tensor product. It is demonstrated how the the approximation is computed in Python and also that sum-factorization is faster to compute than the full tensor product based approach.