Discretization Error Estimation Using Method Of Nearby Problems: One-Dimensional Cases

Abstract

Discretization error is defined as the difference between the solution of the discretized equation and the exact solution of the original partial differential equation. There are two main goals in this study. The first goal is to use of the method of nearby problems to generate exact solutions to realistic problems so that we can asses the performance of discretization error estimators can be assessed. The second goal is to develop and use method of nearby problems itself as an error estimator. Different polynomial curve fitting techniques are examined and fifth-order Hermite splines are identified as the best approach for the method of nearby problems. Steady-state Burgers equation and a modified form of Burgers equation are used as test cases. The analytical curve fits are then the exact solution to a problem nearby the original problem. Results are presented for Burgers equation corresponding to a viscous shock wave for Reynolds numbers of 8 and 64, as well as for a modified version of Burgers equation with a variable viscosity at a nominal Reynolds number of 64. Various discretization error estimators are evaluated for the original Burgers equation, the nearby problem, and the modified version of Burgers equation which includes a nonlinear viscosity term. It is also observed that the method of nearby problems itself performs well as a discretization error estimator even on coarse meshes.