Error transport equations implementation for discontinuous Galerkin methods

Abstract

The procedure for performing higher-order discretization error estimation for Discontinuous Galerkin (DG) methods using Error Transport Equations (ETE) is presented in this work. The results obtained using the linearized ETE and the nonlinear ETE are compared for several steady state inviscid test cases containing completely smooth cases and cases with singularities or discontinuities. Prior work with the linearized version of ETE achieved improved results by using a relinearization process. This process is generalized to the nonlinear version of ETE and is more appropriately termed iterative correction when applied to both the linear and nonlinear versions. For smooth test cases, the discretization error estimates show expected order of accuracy which is higher than the order of accuracy of the primal solution. This holds for both the linearized and the nonlinear ETE. Discontinuities and singularities in the solution cause trouble in the discretization error estimation process. Discussion is made on the behavior of the ETE error estimates for non-smooth test cases to figure out potential solutions to the existing problems.