Obtaining and Verifying High-Order Unstructured Finite Volume Solutions to the Euler Equations

摘要

Interest in high-order unstructured-mesh finite volume methods continues to grow, as researchers are attracted both to the possibilities of high-accuracy solutions on coarse meshes and to applications for which high-order accuracy is all but mandatory, including large-eddy simulations for complex geometries. A major obstacle for those interested in developing a high-order solver, however, is that there are many more details that must be exactly right for a nominally-high-order scheme to be genuinely-high-order than for a second-order scheme to be second-order. In many cases, these details are either omitted or glossed over in the literature. This paper attempts to close this gap by illuminating a number of obscure, or often overlooked, aspects of correctly implementing a high-order-accurate unstructured-mesh finite volume solver. We discuss reconstruction, flux integration, curved-boundary treatment, and demonstrating order of accuracy. In each case, we describe test cases that are appropriate for confirming the correct behavior of a high-order code, as well as demonstrating some of the common pitfalls in implementation and showing their effects. We hope that this paper will serve other researchers as a practical guide to creating their own high-order-accurate solvers.