Fifth-order finite volume WENO in general orthogonally-curvilinear coordinates

Abstract

High order reconstruction in the finite volume (FV) approach is achieved by a more fundamental form of the fifth order WENO reconstruction in the framework of orthogonally-curvilinear coordinates, for solving the hyperbolic conservation equations. The derivation employs a piecewise parabolic polynomial approximation to the zone averaged values to reconstruct the right, middle, and left interface values. The grid dependent linear weights of the WENO are recovered by inverting a Vandermode-like linear system of equations with spatially varying coefficients. A scheme for calculating the linear weights, optimal weights, and smoothness indicator on a regularly- and 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, the procedures for the source term integration and extension to multi-dimensions are proposed. Analytical values of the linear and optimal weights, and also the weights required for the source term integration and flux averaging, are provided for a regularly-spaced grid in Cartesian, cylindrical, and spherical coordinates. Conventional fifth order WENO reconstruction for the regularly-spaced grids in the Cartesian coordinates can be fully recovered in the case of limiting curvature. The fifth order finite volume WENO-C (orthogonally-curvilinear version of WENO) reconstruction scheme is tested for several 1D and 2D benchmark test cases involving smooth and discontinuous flows in cylindrical and spherical coordinates.