Mesh Adaptation for Computational Fluid Dynamics, Volume 2

8Third-Order Unsteady Adaptation

The quest for efficiency leads to combine mesh adaptation with higher order accuracy. This chapter presents a first attempt toward such a combination. A third-order accurate Central Essentially Non-Oscillatory (CENO) approximation is chosen for the demonstration. An a priori error analysis is developed. Then, using a least square projection of the error to a metric-based error reduces the mesh adaptation goal-oriented problem to the optimization of a metric. The numerical illustration concerns nonlinear sound propagation.

8.1. Introduction

Second-order mesh-adaptative approaches bring a good level of safety in the obtention of converging approximate solutions. But these solutions finally obtained are in best case second-order converging to the exact solution. On the other side, high-order solutions are much more efficient when they converge at high order, but often, convergence at high order is not obtained, either because mesh is not sufficient, or because the solution is singular or has stiff variations. Therefore, the Graal for getting accuracy and safety is to combine mesh adaptation and higher order approximations. In order to present a method combining both, we shall choose a higher order approximation. The recent investigations with unstructured meshes concern mainly discontinuous Galerkin (DG) approximations. DG is quite different (data structures, etc.) from the approximations considered in this book. Also, we are interested by singularities, ...