A general positivity-preserving algorithm for implicit high-order finite volume schemes solving the Euler and Navier-Stokes equations

TL;DR

This paper introduces a novel positivity-preserving algorithm for implicit high-order finite volume schemes solving Euler and Navier-Stokes equations, which maintains positivity without restrictive assumptions or reducing time steps.

Contribution

The proposed algorithm is based on asymptotic analysis near vacuum points, does not rely on low-order schemes, and can be integrated into implicit dual time-stepping methods.

Findings

Successfully preserves positive density and pressure in all test cases.

Does not require reduction of time step size for stability.

Maintains accuracy while preserving positivity.

Abstract

This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes solving Euler and Navier-Stokes equations. Previous positivity-preserving algorithms are mainly based on mathematical analyses, being highly dependent on the existence of low-order positivity-preserving numerical schemes for specific governing equations. This dependency poses serious restrictions on extending these algorithms to temporally implicit schemes, since it is difficult to know if a low-order implicit scheme is positivity-preserving. The present positivity-preserving algorithm is based on an asymptotic analysis of the solutions near local vacuum minimum points. The asymptotic analysis shows that the solutions decay exponentially with time to maintain non-negative density and pressure at a local vacuum minimum point. In its neighborhood, the exponential evolution leads to a modification of the linear evolution process, which can be modelled by a direct correction of the linear residual to ensure positivity. This correction however destroys the conservation of the numerical scheme. Therefore, a second correction procedure is proposed to recover conservation. The proposed positivity-preserving algorithm is considerably less restrictive than existing algorithms. It does not rely on the existence of low-order positivity-preserving baseline schemes and the convex decomposition of volume integrals of flow quantities. It does not need to reduce the time step size for maintaining the stability either. Furthermore, it can be implemented iteratively in the implicit dual time-stepping schemes to preserve positivity of the intermediate and converged states of the sub-iterations. It is proved that the present positivity-preserving algorithm is accuracy-preserving. Numerical results demonstrate that the proposed algorithm preserves the positive density and pressure in all test cases.