On The Numerical Accuracy of The Discontinuous Galerkin Method In The Presence of Right-Hand Side Discontinuities

The Discontinuous Galerkin Method (DGM) has been widely used since the 1980s due to its ability to provide accurate high-order (larger than 2) numerical solutions of partial differential equations (PDEs) modelling convection dominated problems and nonlinear ordinary differential equations (ODEs). This work aims to study two effects that can break the high-order numerical accuracy of the DGM: i) the use of finite precision arithmetic, and ii) the existence of discontinuous right-hand sides (source terms). These types of discontinuities occur in many applications in computational science and engineering. To this end, a time-dependent optimal control problem (ODE system) and the shallow water equations (convection-dominated PDE problem) have been studied. The results show that when the discontinuity is inside an element, the numerical accuracy of the solution decreases significantly and is independent of the degree of the polynomial approximation used in the computations. The results also show that this conclusion is independent of the arithmetic precision used in the computations. It is also shown that the number of digits used in the computations, as for example double precision, limits the achievable order of mesh convergence when increasing the polynomial order. The paper also presents an efficient matrix based implementation of the proposed algorithms.