Fifth order finite volume WENO in general orthogonally - curvilinear coordinates

Abstract

A fifth order finite volume WENO reconstruction scheme is proposed in the framework of orthogonally - curvilinear coordinates for solving hyperbolic conservation equations. The derivation employs a piecewise parabolic polynomial approximation to the zone averaged values (Q¯i) to reconstruct the right (qi+), middle (qiM), and left (qi−) interface values. The grid dependent linear weights of the WENO are recovered by inverting a Vandermonde - like linear system of equations with spatially varying coefficients. A scheme for calculating the linear weights, optimal weights, and smoothness indicator on a regularly - /irregularly - spaced grid in orthogonally - curvilinear coordinates is proposed. A grid independent relation for evaluating the smoothness indicator is derived from the basic definition. Finally, a computationally efficient extension to multi - dimensions is proposed along with the procedures for flux and source term integrations. Analytical values of the linear weights, optimal weights, and weights for flux and source term integrations are provided for a regularly - spaced grid in Cartesian, cylindrical, and spherical coordinates. Conventional fifth order WENO - JS can be fully recovered in the case of limiting curvature (R → ∞). The fifth order finite volume WENO - C (orthogonally - curvilinear version of WENO) reconstruction scheme is tested for several 1D and 2D benchmark tests involving smooth and discontinuous flows in cylindrical and spherical coordinates.