Enhanced Accuracy for Finite-Volume and Discontinuous Galerkin Schemes via Non-intrusive Corrections

Abstract

Finite volume and discontinuous Galerkin methods are powerful computational tools for the solution of systems of conservation laws as the Navier Stokes equations. This is due to the fact that they allow piecewise continuous approximations, which turned out to be more robust especially in under-resolved regions or near shock waves. The idea of this paper is to apply an a posteriori post-processing of a steady state solution of a finite volume or a discontinuous Galerkin scheme. The approximation, which consists in every grid cell of a polynomial of degree N, is shifted to polynomials of degree M by reconstruction. The improved approximate solution is inserted into a higher-order approximation to estimate the local discretization error of the obtained solution. This estimated local discretization error of the basic scheme is subtracted from the right hand side of the basic scheme. A new steady state solution is calculated by the modified basic scheme. Iteratively applied, commutes the defect correction the approximation to a steady state solution of higher-order accuracy. For the correction one only needs the inversion of the basic lower-order scheme within an iteration loop. The modification of the basic scheme is non-intrusive and restricted to a change of the right hand side.