Lilia Krivodonova: Accuracy, Linear Stability and Superconvergence of the Discontinuous Galerkin Methods

In this talk we will derive analytical expressions for the eigenvalues of the discontinuous Galerkin (DG) spatial discretization matrix. We will show that the eigenvalues are defined by the [p/p+1] sub-diagonal Pade approximant of the exponential function, where p is the degree of the finite element basis. We then will demonstrate that these eigenvalues are responsible for both (2p+2) superaccuracy in dispersion and dissipation errors and the restrictive, linearly growing with the order of approximation, CFL number of the DG schemes. We will also relate them to the point-wise superconvergence of the DG methods. We will use these results to devise a family of methods that retain the attractive properties of the DGM such as high-order spatial accuracy and locality of approximation but have a lesser restriction on a stable time step.