A High-Order Modified Finite Volume WENO Method on 3D Cartesian Grids

TL;DR

This paper develops sixth-order accurate conversion formulas for a high-order finite volume WENO method on 3D Cartesian grids, enhancing accuracy and robustness for smooth and shock-containing nonlinear problems.

Contribution

It introduces sixth-order accurate face-value conversion formulas for 3D Cartesian grids, improving the existing finite volume WENO method's accuracy and robustness.

Findings

Achieves sixth-order accuracy in face-value conversions.

Maintains high-order accuracy for smooth nonlinear problems.

Robustly handles problems with strong shocks.

Abstract

The modified dimension-by-dimension finite volume (FV) WENO method on Cartesian grids proposed by Buchm\"{u}ller and Helzel can retain the full order of accuracy of the one-dimensional WENO reconstruction and requires only one flux computation per interface. The high-order accurate conversion between face-averaged values and face-center point values is the main ingredient of this method. In this paper, we derive sixth-order accurate conversion formulas on three-dimensional Cartesian grids. It is shown that the resulting modified FV WENO method is efficient and high-order accurate when applied to smooth nonlinear multidimensional problems, and is robust for calculating non-smooth nonlinear problems with strong shocks..