This paper presents a general positivity-preserving algorithm for implicit high-order finite volume schemes that solve compressible Euler and Navier-Stokes equations to ensure the positivity of density and internal energy (or pressure). Previous positivity-preserving algorithms are mainly based on the slope limiting or flux limiting technique, which rely on the existence of low-order positivity-preserving schemes. 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. In the present paper, a new positivity-preserving algorithm is proposed in terms of the flux correction technique. And the factors of the flux correction are determined by a residual correction procedure. For a finite volume scheme that is capable of achieving a converged solution, we show that the correction factors are in the order of unity with additional high-order terms corresponding to the spatial and temporal rates of convergence. Therefore, the proposed positivity-preserving algorithm is accuracy-reserving and asymptotically consistent. The notable advantage of this method is that it does not rely on the existence of low-order positivity-preserving baseline schemes. Therefore, it can be applied to the implicit schemes solving Euler and especially Navier-Stokes equations. In the present paper, the proposed technique is applied to an implicit dual time-stepping finite volume scheme with temporal second-order and spatial high-order accuracy. The present positivity-preserving algorithm is implemented in an iterative manner to ensure that the dual time-stepping iteration will converge to the positivity-preserving solution. Another similar correction technique is also proposed to ensure that the solution remains positivity-preserving at each sub-iteration. Numerical results demonstrate that the proposed algorithm preserves positive density and internal energy in all test cases and significantly improves the robustness of the numerical schemes.

中文翻译：

---

求解欧拉和纳维-斯托克斯方程的隐式高阶有限体积格式的通用正性保持算法

本文提出了一种隐式高阶有限体积格式的通用正性保持算法，该算法求解可压缩欧拉和纳维-斯托克斯方程，以确保密度和内能（或压力）的正性。以往的保正算法主要基于斜率限制或通量限制技术，其依赖于低阶保正方案的存在。这种依赖性对将这些算法扩展到时间隐式方案提出了严重的限制，因为很难知道低阶隐式方案是否是正性保持的。本文在通量校正技术方面提出了一种新的正性保持算法。通量校正的因子由剩余校正程序确定。对于能够实现收敛解的有限体积方案，我们表明校正因子与对应于空间和时间收敛率的附加高阶项处于统一的顺序。因此，所提出的保正算法是保留精度且渐近一致的。该方法的显着优点是它不依赖于低阶正性保留基线方案的存在。因此，它可以应用于求解欧拉方程，特别是纳维-斯托克斯方程的隐式格式。在本文中，所提出的技术应用于具有时间二阶和空间高阶精度的隐式双时间步长有限体积方案。当前的正性保留算法以迭代方式实现，以确保双时间步长迭代将收敛到正性保留解。还提出了另一种类似的校正技术，以确保解在每次子迭代时保持正性保留。数值结果表明，该算法在所有测试用例中都保留了正密度和内能，并显着提高了数值方案的鲁棒性。