Superconvergence of discontinuous Galerkin method for linear hyperbolic equations

Abstract

This thesis is concerned with the investigation of the superconvergence of the Discontinuous Method for linear conservation laws. We use Fourier analysis to study the superconvergence of the semi-discrete discontinuous Galerkin method for scalar linear advection equations in one spatial dimension. We provide the error bounds and asymptotic errors for initial di erent initial discretizations. For the pedagogical purpose, the errors are computed in two di erent ways. In the rst approach, we compute the di erence between the numerical solution and a special interpolation of the exact solution, and show that it consists of an asymptotic error of order 2k + 1 (where k is the order of polynomial approximation) and a transient error of lower order. In the second approach, we compute the error directly by decomposing it into physical and nonphysical modes, and obtain agreement with the rst approach. We then extend the analysis to vector conservation laws, solved using the Lax-Friedrichs ux. We prove that the superconvergence holds with the same order. The error bounds and asymptotic errors are demonstrated by various numerical experiments for scalar and vector advection equations.