Very high-order methods for 3D arbitrary unstructured grids

Abstract

Understanding the motion of fluids is crucial for the development and analysis of new designs and processes in science and engineering. Unstructured meshes are used in this context since they allow the analysis of the behaviour of complicated geometries and configurations that characterise the designs of engineering structures today. The existing numerical methods developed for unstructured meshes suffer from poor computational efficiency, and their applicability is not universal for any type of unstructured meshes. High-resolution high-order accurate numerical methods are required for obtaining a reasonable guarantee of physically meaningful results and to be able to accurately resolve complicated flow phenomena that occur in a number of processes, such as resolving turbulent flows, for direct numerical simulation of Navier-Stokes equations, acoustics etc. The aim of this research project is to establish and implement universal, high-resolution, very high-order, non-oscillatory finite-volume methods for 3D unstructured meshes. A new class of linear and WENO schemes of very high-order of accuracy (5 th ) has been developed. The key element of this approach is a high-order reconstruction process that can be applied to any type of meshes. The linear schemes which are suited for problems with smooth solutions, employ a single reconstruction polynomial obtained from a close spatial proximity. In the WENO schemes the reconstruction polynomials, arising from different topological regions, are non-linearly combined to provide high-order of accuracy and shock capturing features. The performance of the developed schemes in terms of accuracy, non-oscillatory behaviour and flexibility to handle any type of 3D unstructured meshes has been assessed in a series of test problems. The linear and WENO schemes presented achieve very high-order of accuracy (5 th ). This is the first class of WENO schemes in the finite volume context that possess highorder of accuracy and robust non-oscillatory behaviour for any type of unstructured meshes. The schemes have been employed in a newly developed 3D unstructured solver (UCNS3D). UCNS3D utilises unstructured grids consisted of tetrahedrals, pyramids, prisms and hexahedral elements and has been parallelised using the MPI framework. The high parallel efficiency achieved enables the large scale computations required for the analysis of new designs and processes in science and engineering.