A recipe to detect the error in discretization schemes

Abstract

AbstractThe necessity for a reliable measure of the discretization error arises in adaptive mesh refinement and in moving mesh adaptation. The present work discusses a detector of the discretization error based on the interpolation reconstruction of the operators. The technique presented here is named operator recovery error source detector (ORESD). Its main features are: First, the technique is based on the operators being discretized and does not require any user intervention or any a priori knowledge of the solution or its properties. Second, the ORESD is an a posteriori error indicator, but it is shown to be consistent with the a priori error provided by the modified equation approach. Third, the technique is based on the operators being solved and is tailored to the specific problem at hand. Four, the technique is simple and is based on a small stencil, resulting in a very inexpensive error detection.In the present work, the ORESD is derived and applied to two tutorial examples: divergence and gradient. With the aid of the two examples and using the general derivation, the ORESD is then applied to the gas dynamics equations. Two benchmarks are used to test the performance. First, a shock tube problem is solved (Sod's benchmark) in a Lagrangian and in a Eulerian frame. Second, the Colella's wedge problem is solved using CLAWPACK. Finally, the ORESD is applied to the 2D Poisson equation on a uniform and on a non‐uniform grid to test the application to elliptic problems. In all examples the operator recovery error source detector succeeds in detecting the real sources of error. Copyright © 2004 John Wiley & Sons, Ltd.