Accuracy of Discretization Error Estimation by the Error Transport Equation on Unstructured Meshes - Nonlinear Systems of Equations

Abstract

In this paper, we perform a numerical estimation of discretization error using the error transport equation, derived from the primal PDE. The viability of using this method for obtaining higher order estimates for unstructured finite-volume discretizations of scalar linear and nonlinear scalar PDEs has previously been demonstrated, and here we examine how this extends to steady state solutions to the Euler equations, a nonlinear system of PDEs. Considerations for the error transport equation with and without linearization were made. Comparisons of results show that using the fully nonlinear form has verifiable properties as well as being superior in accuracy of the error estimate in some situations, although the Newton linearization can be adequate in others. The major results for 1D and 2D test cases were consistent with scalar problems. With arbitrary choices of discretization orders for the primal and error PDEs and residual source term, the error estimate obtained is in general not sharp and converges to the exact error at the same order as the primal discretization. However, using a discretization scheme where the source term for the error equation is the residual based on a reconstruction of the converged primal solution that is the same order as the error equation discretization leads to a sharp, high order estimate compared to other combinations. Therefore, we demonstrate that there are nominal accuracy combinations for discretizing the primal and error equations, and evaluating the residual source term, that require more computational work but are actually less accurate asymptotically in obtaining an estimate of error, which are choices that one should never make in practice. In addition, some results for the runtime costs are obtained for evaluating the feasibility of applying this error estimation approach compared to higher order primal discretizations.