Three-dimensional high-order finite-volume method based on compact WENO reconstruction with hybrid unstructured grids

Abstract

In the present work, a three-dimensional (3D) high-order finite-volume method, based on the compact simple weighted essentially non-oscillatory (WENO) reconstruction for hybrid unstructured grids, is developed. For each control cell, its stencils only involve neighboring cells. By ensuring that the various orders of the derivatives of the conservative variables are conserved over the stencils, an over-determined system of equations is formed, and the polynomials corresponding to each stencil are obtained by solving this system of equations using the least-squares method. The artificially determined weight coefficients, and the smoothing factors based on the smoothness of the flow field, are used to determine the nonlinear weights, which nonlinearly combine the polynomials of each stencil to obtain the final high-order reconstruction polynomial. Consequently, the conservative variables on the interface can be interpolated and the sphere-function based discrete gas-kinetic scheme (DGKS) is used to locally solve the Boltzmann equation. Lastly, the fluxes are calculated using the mesoscopic perspective based on the moment relations. The present algorithm has the following advantages: the stencils only involve the neighboring cells of each control volume, making the algorithm compact and simpler, especially for the operation of the boundary cells; the unknown coefficients are solved by the least-squares method to avoid the coefficient matrix being ill-conditioned; the simple WENO scheme artificially chooses the value to guarantee positive weights, in addition, the algorithm can be applied to 3D hybrid unstructured grids, making it more adaptable. In terms of flux calculation, DGKS not only inherits the advantages of the traditional gas-kinetic schemes (GKS) but also greatly simplifies the flux calculation expressions. Several cases are simulated to verify the correctness and robustness of the algorithm, and the accuracy test shows the algorithm is third-order accurate.