Symmetry-preserving enforcement of low-dissipation method based on boundary variation diminishing principle

TL;DR

This paper addresses symmetry-breaking issues in high-order shock-capturing schemes for Euler equations, proposing modifications to ensure symmetry preservation by mitigating round-off errors in finite volume computations.

Contribution

It introduces new techniques to eliminate symmetry-breaking causes in P4T2-BVD schemes, ensuring symmetric solutions in high-resolution flow simulations.

Findings

Proposed modifications successfully preserve symmetry in benchmark tests.

Enhanced scheme robustness against round-off errors.

Achieved perfect symmetric solutions in numerical experiments.

Abstract

A class of high-order shock-capturing schemes, P$_n$T$_m$-BVD (Deng et al., J. Comp. Phys., 386:323-349, 2019; Comput. & Fluids, 200:104433, 2020.) schemes, have been devised to solve the Euler equations with substantially reduced numerical dissipation, which enable high-resolution simulations to resolve flow structures of wider range scales. In such simulations with low dissipation, errors of round-off level might grow and contaminate the numerical solutions. A typical example of such problems is the loss of symmetry in the numerical solutions for physical problems of symmetric configurations even if the schemes are mathematically in line with the symmetry rules. In this study, the mechanisms of symmetry-breaking in a finite volume framework with the P$_4$T$_2$-BVD reconstruction scheme are thoroughly examined. Particular attention has been paid to remove the possible causes due to the lack of associativity in floating-point arithmetic which is associated with round-off errors. Modifications and new techniques are proposed to completely remove the possible causes for symmetry breaking in different components of the P$_4$T$_2$-BVD finite volume solver. Benchmark tests that have symmetric solution structures are used to verify the proposed methods. The numerical results demonstrate the perfect symmetric solution structures.