We propose a stable scheme to solve numerically the Cahn–Hilliard–Hele–Shaw system in three-dimensional space. In the proposed scheme, we discretize the space and time derivative terms by combining with backward differentiation formula, which turns out to be both second-order accurate in space and time. Using this method, a set of linear elliptic equations are solved instead of the complicated and high-order nonlinear equations. We prove that our proposed scheme is uniquely solvable. We use a linear multigrid solver, which is fast and convergent, to solve the resulting discrete system. The numerical tests indicate that our scheme can use a large time step. The accuracy and other capability of the proposed algorithm are demonstrated by various computational results.

我们提出了一种稳定的方案来在三维空间中数值求解Cahn–Hilliard–Hele–Shaw系统。在所提出的方案中，我们结合后向微分公式离散化了时空导数项，结果证明它们在时空上都是二阶精确的。使用这种方法，可以求解一组线性椭圆方程，而不是复杂的高阶非线性方程。我们证明了我们提出的方案是唯一可解的。我们使用快速且收敛的线性多重网格求解器来求解最终的离散系统。数值测试表明，我们的方案可以使用较大的时间步长。各种计算结果证明了所提算法的准确性和其他能力。