Fifth order finite volume WENO in general orthogonally-curvilinear coordinates

TL;DR

This paper develops a fifth order finite volume WENO reconstruction method tailored for orthogonally-curvilinear coordinates, enabling high accuracy in solving hyperbolic conservation equations on complex grids.

Contribution

It introduces a fundamental derivation of WENO reconstruction in curvilinear coordinates, including new formulas for weights and smoothness indicators applicable to various coordinate systems.

Findings

Achieves high-order accuracy in curvilinear coordinates

Successfully tested on 1D and 2D benchmark problems

Provides analytical weight formulas for different coordinate systems

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.