Abstract
In this paper, a family of stencil selection algorithms is presented for WENO schemes on unstructured meshes. The associated freedom of stencil selection for unstructured meshes, in the context of WENO schemes present a plethora of various stencil selection algorithms. The particular focus of this paper is to assess the performance of various stencil selection algorithm, investigate the parameters that dictate their robustness, accuracy and computational efficiency. Ultimately, efficient and robust stencils are pursued that can provide significant savings in computational performance, while retaining the non-oscillatory character of WENO schemes. This is achieved when making the stencil selection algorithms adaptive, based on the quality of the cells for unstructured meshes, that can in turn reduce the computational cost of WENO schemes. For assessing the performance of the developed algorithms well established test problems are employed. These include the least square approximation of polynomial functions, linear advection equation of smooth functions and solid body rotation test problem. Euler and Navier-Stokes equations test problems are also pursued such as the Shu-Osher test problem, the Double Mach Reflection, the supersonic Forward Facing step, the Kelvin-Helmholtz instability, the Taylor-Green Vortex, and the flow past a transonic circular cylinder.
Highlights
Weighted Essentially non-Oscillatory (WENO) schemes have established themselves as a prevalent technique, for providing non-oscillatory properties, to various high-fidelity high-order numerical frameworks suited for unstructured meshes including the Finite Volume (FV) [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and the Discontinuous Galerkin (DG) [15,16,17,18,19,20,21] frameworks
Due to the inherent increased computational cost of the WENO schemes for unstructured meshes, a recent focus has been to reduce their computational cost by employing optimised numerical libraries [22], or using a compact implementation of them called Compact WENO (CWENO) schemes [8,9,10] in order to reduce the size of the directional stencils
Due to the unstructured meshes, and the nature of the WENO schemes in this context, it can be argued that there is a lot of freedom regarding the choice of stencils, since the arbitrariness of the meshes dictates that generic algorithms that can be employed for any mesh are desirable
Summary
Weighted Essentially non-Oscillatory (WENO) schemes have established themselves as a prevalent technique, for providing non-oscillatory properties, to various high-fidelity high-order numerical frameworks suited for unstructured meshes including the Finite Volume (FV) [1,2,3,4,5,6,7,8,9,10,11,12,13,14] and the Discontinuous Galerkin (DG) [15,16,17,18,19,20,21] frameworks. A series of stringent 2D and 3D test-problems including the Linear advection, Euler and Navier-Stokes equations using unstructured meshes of various elements types are considered for assessing the performance of all the developed stencil selection algorithms. The conclusions drawn from the present study are outlined in the last section
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