Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids
Kernel-free boundary integral method for two-phase Stokes equations with discontinuous viscosity on staggered grids
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DOI：
10.2139/ssrn.4353542
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发表时间：
2023-02
期刊：
影响因子：
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通讯作者：
H. Dong;Shuwang Li;W. Ying;Zhongshu Zhao
中科院分区：
文献类型：
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作者：
H. Dong;Shuwang Li;W. Ying;Zhongshu Zhao
不连续的粘度系数使得速度和法向应力的跳跃条件耦合在一起，这给一些常用的数值方法获得精确解带来了巨大的挑战。为了克服这些困难，提出了一种结合改进的标记单元（MAC）方案的无核边界积分（KFBI）方法来解决具有不连续粘度的两相斯托克斯问题。其主要思想是利用边界积分方程将两相斯托克斯问题重新表述为单流体斯托克斯问题，然后通过基于笛卡尔网格的方法间接评估边界积分。由于单流体斯托克斯问题的跳跃条件可以很容易地解耦，因此这里采用改进的MAC方案，并且现有的快速求解器可以适用于所得的线性鞍座系统。计算出的数值解在速度和压力以及速度梯度的离散 $\ell^2$-范数中是二阶精确的，并且在速度及其梯度的最大范数中也是二阶精确的，即使在高对比粘度系数的情况下也是如此，这在数值测试中得到了证明。
A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete $\ell^2$-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests.