Arbitrary high order central non-oscillatory schemes on mixed-element unstructured meshes

Abstract

In this paper we develop a family of very high-order central (up to 6th-order) non-oscillatory schemes for mixed-element unstructured meshes. The schemes are inherently compact in the sense that the central stencils employed are as compact as possible, and that the directional stencils are reduced in size therefore simplifying their implementation. Their key ingredient is the non-linear combination in a CWENO style similar to Dumbser et al [1] of a high-order polynomial arising from a central stencil with lower-order polynomials from directional stencils. Therefore, in smooth regions of the computational domain the optimum order of accuracy is recovered, while in regions of sharp-gradients the larger influence of the reconstructions from the directional stencils suppress the oscillations. It is the compactness of the directional stencils that increases the chances of at least one of them lying in a region with smooth data, that greatly enhances their robustness compared to classical WENO schemes. The two variants developed are CWENO and CWENOZ schemes, and it is the first time that such very-high-order schemes are designed for mixed-element unstructured meshes. We explore the influence of the linear weights in each of the schemes, and assess their performance in terms of accuracy, robustness and computational cost through a series of stringent 2D and 3D test problems. The results obtained demonstrate the improved robustness that the schemes offer, a parameter of paramount importance for and their potential use for industrial-scale engineering applications.