Second Order Convergence of a Modified MAC Scheme for Stokes Interface Problems
Second Order Convergence of a Modified MAC Scheme for Stokes Interface Problems
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DOI：
10.1007/s10915-023-02239-w
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发表时间：
2023-02
影响因子：
2.5
通讯作者：
H. Dong;Zhongshu Zhao;Shuwang Li;W. Ying;Jiwei Zhang
中科院分区：
文献类型：
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作者：
H. Dong;Zhongshu Zhao;Shuwang Li;W. Ying;Jiwei Zhang
Stokes流动方程已成功地应用于模拟具有移动界面的问题。虽然计算方法的基础上著名的MAC计划产生精确的解决方案和数值收敛可以证明使用分辨率的研究，严格的收敛性证明通常限于特定的重新制定和边界条件。本文在有限差分法的框架下，对常粘Stokes界面问题的标记和单元格式进行了严格的误差分析。在不将问题转化为椭圆型偏微分方程的情况下，主要思想是利用离散的Ladyzenskaja-Babuska-Brezzi条件，构造满足离散Stokes方程且在运动界面附近具有至少二阶精度的辅助函数.特别是，该方法，第一次，使一个证明二阶收敛的速度梯度的离散范数，除了速度和压力场。数值实验验证了所需的属性的方法和预期的顺序的精度为二维和三维的例子。
Stokes flow equations have been implemented successfully in practice for simulating problems with moving interfaces. Though computational methods based on the well-known MAC scheme produce accurate solutions and numerical convergence can be demonstrated using a resolution study, the rigorous convergence proofs are usually limited to particular reformulations and boundary conditions. In this paper, a rigorous error analysis of the marker and cell scheme for Stokes interface problems with constant viscosity in the framework of the finite difference method is presented. Without reformulating the problem into elliptic PDEs, the main idea is to use a discrete Ladyzenskaja-Babuska-Brezzi condition and construct auxiliary functions, which satisfy discretized Stokes equations and possess at least second order accuracy in the neighborhood of the moving interface. In particular, the method, for the first time, enables one to prove second order convergence of the velocity gradient in the discrete-norm, in addition to the velocity and pressure fields. Numerical experiments verify the desired properties of the methods and the expected order of accuracy for both two-dimensional and three-dimensional examples.