Improved Accuracy of High-Order WENO Finite Volume Methods on Cartesian Grids

TL;DR

This paper introduces a modification to high-order WENO finite volume methods on Cartesian grids that maintains full spatial accuracy for nonlinear multidimensional conservation laws, demonstrated through tests on gas dynamics.

Contribution

A simple modification to standard WENO methods that preserves the full spatial order of accuracy in multidimensional settings for conservation laws.

Findings

High-order accuracy confirmed for smooth nonlinear problems

Method effectively applied to 2D Euler equations

Extension to 3D and other systems is straightforward

Abstract

We propose a simple modification of standard WENO finite volume methods for Cartesian grids, which retains the full spatial order of accuracy of the one-dimensional discretization when applied to nonlinear multidimensional systems of conservation laws. We derive formulas, which allow us to compute high-order accurate point values of the conserved quantities at grid cell interfaces. Using those point values, we can compute a high-order flux at the center of a grid cell interface. Finally, we use those point values to compute high-order accurate averaged fluxes at cell interfaces as needed by a finite volume method. The method is described in detail for the two-dimensional Euler equations of gas dynamics. An extension to the three-dimensional case as well as to other nonlinear systems of conservation laws in divergence form is straightforward. Furthermore, similar ideas can be used to improve the accuracy of WENO type methods for hyperbolic systems which are not in divergence form. Several test computations confirm the high-order accuracy for smooth nonlinear problems.