The level set method is a common approach for handling moving boundary problems, which allows a moving, irregular surface to be described implicitly on a Cartesian grid. This approach often requires reinitialization of the level set function and extrapolation of fields defined only on the interface. Because many applications in physics and engineering involve calculation of second derivatives of the interface curvature and fourth order derivatives of surface fields, accurate simulations of these problems require high-order methods for reinitialization and extrapolation. Here we build off WENO schemes for Hamilton–Jacobi equations to develop novel sixth-order accurate methods for reinitialization and extrapolation. We present numerical results in three dimensional spaces demonstrating fourth-order accuracy of the interfacial curvature and sixth-order accuracy for the extrapolated surface fields. We then show that the extrapolation scheme can be integrated into the closest point method for surface PDEs and present an example of computing geodesic curves on surfaces.

水平集方法是处理运动边界问题的常用方法，它允许在笛卡尔网格上隐式描述运动的不规则表面。这种方法通常需要重新初始化级别集功能并外推仅在接口上定义的字段。因为在物理学和工程学中的许多应用都涉及界面曲率的二阶导数和表面场的四阶导数的计算，所以对这些问题的精确模拟需要用于重新初始化和外推的高阶方法。在这里，我们为汉密尔顿-雅各比方程建立WENO方案，以开发新颖的六阶精确方法进行重新初始化和外推。我们在三维空间中显示数值结果，表明界面曲率的四阶精度和外推表面场的六阶精度。然后，我们表明可以将外推方案集成到表面PDE的最近点方法中，并提供计算表面上测地线曲线的示例。