A sufficient condition for free-stream preserving in the nonlinear conservative finite difference schemes on curvilinear grids

TL;DR

This paper establishes a sufficient condition for free-stream preservation in nonlinear conservative finite difference schemes on curvilinear grids, ensuring geometric conservation law compliance without altering the original scheme forms.

Contribution

It introduces a general criterion for free-stream preservation, constructs new FP metrics for WENO5 and WENO7, and maintains scheme accuracy with simple modifications.

Findings

FP schemes preserve free-stream condition on curvilinear grids.

The new metrics retain high-order accuracy and robustness.

Validation shows effective resolution of smooth regions and discontinuities.

Abstract

In simulations of compressible flows, the conservative finite difference method (FDM) based on the nonlinear upwind schemes, e.g. WENO5, might violate free-stream preserving (FP), due to the loss of the geometric conservation law (GCL) identity when applied on the curvilinear grids. Although some techniques on FP have been proposed previously, no general rule is given for this issue. In this paper, by rearranging the upwind dissipation of the nonlinear schemes as a combination of sub-stencil reconstructions (taking WENO5 as an example), it can be proved that the upwind dissipation diminishes under the uniform flow condition if the metrics yield an identical value under the same schemes with these reconstructions, making the free-stream condition be preserved. According to this sufficient condition, the novel FP metrics are constructed for WENO5 and WENO7. By this means the original forms of these WENO schemes can be kept. In addition, the accuracy of these schemes can be retained as well with a simple accuracy compensation by replacing the central part fluxes with a high-order one. Various validations indicate that the present FP schemes retain the great capability to resolve the smooth regions accurately and capture the discontinuities robustly.