Stencil selection algorithms for WENO schemes on unstructured meshes

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 nonoscillatory 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. Crown

Application of Central-Weighted Essentially Non-Oscillatory Finite-Volume Interface-Capturing Schemes for Modeling Cavitation Induced by an Underwater Explosion

“…In regions of discontinuous solutions, the reconstructed solution will be mostly influenced from the lower-order polynomials of the directional stencils using the Type 3 directional stencils. The reader is referred to the work of Tsoutsanis [39] for the stencil selection algorithms. ãk are the reconstructed degrees of freedom, and the non-linear weight ω s is defined as such that the non-linear coefficients → s are computed from the linear ones → s as:…”

Application of CWENO Finite-Volume Interface Capturing Schemes for Modeling Cavitation Induced by an Underwater Explosion

“…where ω s correspond to the non-linear weights assigned to each polynomial, and in regions with smooth data ω s ≈ λ s , hence obtaining the high-order approximation from the central stencil, and in regions of discontinuous solutions the reconstructed solution will be mostly influenced from the lower-order polynomials of the directional stencils using the Type 3 directional stencils. The reader is referred to the work of Tsoutsanis [61] for the stencil selection algorithms. ãk are the reconstructed degrees of freedom; and the non-linear weight ω s is defined as: the nonlinear coefficients → s are computed from the linear ones → s as:…”