Fifth-order finite-volume weno on cylindrical grids

Abstract

Fifth-order finite-volume WENO is proposed for a structured grid in cylindrical coordinates. The derivation of linear weights, optimal weights uses a polynomial approximation in a dimension-by-dimension framework, implemented with the local conservation property. Finally, a Vandermonde-like system is obtained, which can be solved for linear weights on both regularly-spaced and irregularly-spaced grids in cylindrical coordinates, where the analytical relations can be derived for the former. In addition, a grid-independent formulation for evaluating the smoothness indicators is derived by minimizing the L2-norm of the derivatives of reconstruction polynomial. The scheme converges to WENO-JS for the limiting case (R →∞). A linear stability analysis of the proposed reconstruction scheme is performed using a 1D scalar advection equation in cylindrical-radial coordinates. Several tests are performed to assess the performance of the proposed scheme. The results indicate that WENO-Curvilinear significantly improves the results when compared with the previous methods.