This paper is concerned with the mean curvature flow, which describes the dynamics of a hypersurface whose normal velocity is determined by local mean curvature. We present a Cartesian grid-based method for solving mean curvature flows in two and three space dimensions. The present method embeds a closed hypersurface into a fixed Cartesian grid and decomposes it into multiple overlapping subsets. For each subset, extra tangential velocities are introduced such that marker points on the hypersurface only moves along grid lines. By utilizing an alternating direction implicit (ADI)-type time integration method, the subsets are evolved alternately by solving scalar parabolic partial differential equations on planar domains. The method removes the stiffness using a semi-implicit scheme and has no high-order stability constraint on time step size. Numerical examples in two and three space dimensions are presented to validate the proposed method.

翻译：本文研究平均曲率流，该流描述了法向速度由局部平均曲率决定的超曲面的动力学。我们提出了一种基于笛卡尔网格的方法，用于求解二维和三维空间中的平均曲率流。该方法将封闭超曲面嵌入固定笛卡尔网格中，并将其分解为多个重叠子集。对于每个子集，引入额外的切向速度，使得超曲面上的标记点仅沿网格线移动。通过利用交替方向隐式（ADI）型时间积分方法，这些子集通过求解平面区域上的标量抛物型偏微分方程交替演化。该方法采用半隐式格式消除刚性问题，且对时间步长无高阶稳定性约束。我们给出了二维和三维空间中的数值算例，以验证所提出方法的有效性。