Superconvergence of the space-time discontinuous Galerkin method for linear nonhomogeneous hyperbolic equations

Abstract

In this study, we discuss the superconvergence of the space-time discontinuous Galerkin method for the first-order linear nonhomogeneous hyperbolic equation. By using the local differential projection method to construct comparison function, we prove that the numerical solution is $$(2n+1)$$ -th order superconvergent at the downwind-biased Radau points in the discrete $$L^2$$ -norm. As a by-product, we obtain a point-wise superconvergence with order $$2n+\frac{1}{2}$$ in vertices. We also find that, in order to obtain these superconvergence results, the source integral term has to be approximated by $$(n+1)$$ -point Radau-quadrature rule. Numerical results are presented to verify our theoretical findings.