A sixth-order finite difference scheme with the minimized dispersion and adaptive dissipation for solving compressible flow

TL;DR

This paper introduces a sixth-order finite difference scheme with minimized dispersion and adaptive dissipation, enhancing high-fidelity simulations of compressible flow with broadband scales.

Contribution

The paper develops a novel sixth-order scheme with a modified scale sensor for minimized dispersion and adaptive dissipation, improving accuracy over previous methods.

Findings

Demonstrates high resolution in benchmark tests with broadband scales.

Achieves minimized dispersion and adaptive dissipation in simulations.

Outperforms traditional schemes in preserving flow details.

Abstract

The dispersion and dissipation properties of a scheme are important to realize high-fidelity simulations of the compressible flow, especially the cases with broadband length scales. It has been recognized that the minimization of dispersion error is an effective method to improve the precision. In addition, the proper dissipation of a scheme is important to restrain the non-physics oscillations and reserve details of flows simultaneously. The authors have previously proposed a scale sensor to adjust the numerical dissipation of a fourth-order finite difference scheme according to the local scale of the flow. In this paper, the scale sensor is further modified for the sixth-order finite difference scheme to achieve minimized dispersion and adaptive dissipation properties. Firstly, the scale sensor quantifies the local length scale of the numerical solution as the effective scaled wavenumber. Then, the dispersion-dissipation condition is used to construct the relationship between the dissipation/dispersion parameter and the effective scaled wavenumber. Therefore, a sixth-order finite difference scheme with minimized dispersion and adaptive dissipation (MDAD6th) is proposed. Several benchmark test cases with broadband length scales are presented to clarify the high resolution of the new scheme.