Error estimates and post-processing of local discontinuous Galerkin method for Schrödinger equations

Abstract

In this paper, we present L 2 and negative-order norm estimates for the local discontinuous Galerkin (LDG) method solving variable coefficient Schrödinger equations. For these special solutions the LDG method exhibits “hidden accuracy”, and we are able to extract it through the use of a convolution kernel that is composed of a linear combination of B-splines. This technical was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving (SIAC) filter. We demonstrate that it is possible to extend the SIAC filter on Schrödinger equations. When polynomials of degree k are used, we can prove theoretically the LDG method solutions are of order k + 1 , whereas the post-processed solutions that convolution with the SIAC filter are of order at least 2 k . Additionally, we present numerical results to confirm that the accuracy of LDG solutions can be improved from O ( h k + 1 ) to at least O ( h 2 k + 1 ) for Schrödinger equations by using alternating numerical fluxes.