In this paper, we present a semi-implicit numerical solver for a first order hyperbolic formulation of two-phase flow with surface tension and viscosity. The numerical method addresses several complexities presented by the PDE system in consideration: (i) The presence of involution constraints of curl type in the governing equations requires explicit enforcement of the zero-curl property of certain vector fields (an interface field and a distortion field); the problem is eliminated entirely by adopting a set of compatible curl and gradient discrete differential operators on a staggered grid, allowing to preserve the Schwartz identity of cross-derivatives exactly. (ii) The evolution equations feature highly nonlinear stiff algebraic source terms which are used for the description of viscous interactions as emergent behaviour of an elasto-plastic solid in the stiff strain relaxation limit; such source terms are reliably integrated with an efficient semi-analytical technique. (iii) In the low-Mach number regime, standard explicit Godunov-type schemes lose efficiency and accuracy; the issue is addressed by means of a simple semi-implicit, pressure-based, split treatment of acoustic and non-acoustic waves, again using staggered grids that recover the implicit solution for a single scalar field (the pressure) through a sequence of symmetric-positive definite linear systems that can be efficiently solved via the conjugate gradient method.

翻译：在本文中,我们提出了一个半隐含数字求解器,用于对表面紧张和粘度的两阶段流动进行双向双向双向双向双向配制。数字方法处理PDE系统在考虑时提出的若干复杂问题:(一) 在治理方程中存在卷律类型的进化限制,要求明确执行某些矢量字段的零线属性(一个界面字段和一个扭曲字段);通过在交错格格网格上采用一套兼容的卷轴和梯度离散操作器来完全解决问题,从而能够准确保存交叉诊断器的Swartz特性。 (二) 进化方程式具有高度非线性硬代数源词的特点,这些词用于描述在严格压力放松限度内刻的弹性固态固态固态固态固态;这种源术语与有效的半分析技术可靠地融合在一起。 (三) 在低兆位数制度中,标准明确的哥杜诺夫型计划会失去效率和准确性;这一问题的解决方式是使用简单的半隐性半闭定态的直径直径直径的直径直径直径直径直径直径直径直的平方的直径直径直径直径直径直径直径直的直的直径直方的直径直压方法,即通过直地从一个直的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流的平流处理。