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This paper presents a general positivity-preserving algorithm for implicithigh-order finite volume schemes solving Euler and Navier-Stokes equations.Previous positivity-preserving algorithms are mainly based on mathematicalanalyses, being highly dependent on the existence of low-orderpositivity-preserving numerical schemes for specific governing equations. Thisdependency poses serious restrictions on extending these algorithms totemporally implicit schemes, since it is difficult to know if a low-orderimplicit scheme is positivity-preserving. The present positivity-preservingalgorithm is based on an asymptotic analysis of the solutions near local vacuumminimum points. The asymptotic analysis shows that the solutions decayexponentially with time to maintain non-negative density and pressure at alocal vacuum minimum point. In its neighborhood, the exponential evolutionleads to a modification of the linear evolution process, which can be modelledby a direct correction of the linear residual to ensure positivity. Thiscorrection 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 restrictivethan existing algorithms. It does not rely on the existence of low-orderpositivity-preserving baseline schemes and the convex decomposition of volumeintegrals of flow quantities. It does not need to reduce the time step size formaintaining the stability either. Furthermore, it can be implementediteratively in the implicit dual time-stepping schemes to preserve positivityof the intermediate and converged states of the sub-iterations. It is provedthat the present positivity-preserving algorithm is accuracy-preserving.Numerical results demonstrate that the proposed algorithm preserves thepositive density and pressure in all test cases.

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