SIAM Journal on Scientific Computing, Volume 43, Issue 4, Page B1029-B1055, January 2021.
The positivity preservation or hyperbolicity preservation of numerical schemes is essential for robust computations of compressible two-fluid models. Due to the quasi-conservative (nonconservative) nature of the compressible two-fluid model and use of the stiffened-gas equation of state that allows liquid-gas modeling, the analysis of the positivity preservation is significantly more challenging than that of the single-fluid case based on Euler's equations of gas dynamics with ideal gases. We present a detailed, thorough analysis of positivity preservation of widely accepted first-order schemes for the quasi-conservative compressible two-fluid model. The analysis is based on whether material parameters appearing in the stiffened-gas equation of state satisfy a concavity condition. When the concavity condition is satisfied, a proof of positivity preservation of the numerical scheme is presented. When the concavity condition is not satisfied, it is discovered that the positivity preservation is not necessarily ensured, and examples that lead to loss of positivity (or loss of hyperbolicity) are derived. The analysis and numerical verifications are carried out in both one and higher space dimensions. The positivity-preservation proof provides rigorous justification that the first-order scheme when yielding hyperbolicity-preserving solutions can be used as a basis for the design of a high-order hyperbolicity-preserving scheme for the compressible two-fluid model. On the other hand, the demonstration of the lack of positivity in cases where the concavity property is not satisfied by the fluid parameters reveals the possible loss of stability and robustness of the widely accepted first-order scheme for the quasi-conservative compressible two-fluid model in some extreme problems.

中文翻译：

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准保守可压缩双材料模型一阶方案的正性保持

SIAM 科学计算杂志，第 43 卷，第 4 期，第 B1029-B1055 页，2021 年 1 月。
数值方案的正性保持或双曲线保持对于可压缩双流体模型的稳健计算至关重要。由于可压缩二流体模型的准守恒（非守恒）性质以及允许液态气建模的硬化气体状态方程的使用，正性保持的分析比单流体模型更具挑战性。基于欧拉气体动力学方程的流体情况与理想气体。我们对准保守可压缩二流体模型的广泛接受的一阶方案的正性保持进行了详细、透彻的分析。该分析基于出现在硬化气体状态方程中的材料参数是否满足凹度条件。当满足凹度条件时，给出了数值方案的正性保持的证明。当不满足凹度条件时，发现不一定能保证正性保持，并推导出导致正性损失（或双曲线损失）的例子。分析和数值验证在一维和更高的空间维度上进行。正性保持证明提供了严格的证明，即在产生双曲线保持解时的一阶方案可用作设计可压缩二流体模型的高阶双曲线保持方案的基础。另一方面，