Mesh Adaptation for Computational Fluid Dynamics, Volume 1

6Multiscale Convergence and Certification in CFD

While the previous chapters explain how the mesh adaptation loop replaces a mesh by a new mesh, with the same complexity/number of nodes, but optimal in some sense, this chapter studies the search of a sufficiently mesh-converged computation. The mesh-convergence algorithm is an outer loop containing the mesh adaptation algorithm. As an example, convergence is analyzed for an academic case and the new algorithm is demonstrated with the flow around a supersonic aircraft shape.

6.1. Introduction

The previous chapter proposed an algorithm for replacing a given mesh by an adapted mesh with a prescribed complexity. The accuracy is better than with a non-adaptive mesh but the user cannot evaluate if the result is of sufficient accuracy or not. The purpose of this chapter is to use the previous chapters outputs in order to build a mesh adaptation method for PDEs producing certifiable results. Results are certified via mesh convergence measurements. Before doing this, we would like to examine what are the needs for an increased accuracy, in particular in compressible CFD models. While numerical methods and computers show impressive improvements, the numerical quality of CFD computations remains a delicate problem (see, for example, Hutton and Casey 2001). In particular, estimating approximation errors is one of today’s most important challenges in CFD (Rumsey et al. 2018; Morrison 2022). Existing error estimates are progessively more ...