It is well-known that the standard level set advection equation does not preserve the signed distance property, which is a desirable property for the level set function representing a moving interface. Therefore, reinitialization or redistancing methods are frequently applied to restore the signed distance property while keeping the zero-contour fixed. As an alternative approach to these methods, we introduce a modified level set advection equation that intrinsically preserves the norm of the gradient at the interface, i.e. the local signed distance property. Mathematically, this is achieved by introducing a carefully chosen source term being proportional to the local rate of interfacial area generation. The introduction of the source term turns the problem into a non-linear one. However, we show that by discretizing the source term explicitly in time, it is sufficient to solve a linear equation in each time step. Notably, without further adjustment, the method works in the case of a moving contact line. This is a major advantage since redistancing is known to be an issue when contact lines are involved. We provide a first implementation of the method in a simple first-order upwind scheme in both two and three spatial dimensions.

翻译：众所周知，标准水平集平流方程不保持符号距离性质——而这正是表示移动界面的水平集函数所期望的特性。因此，通常需要应用重新初始化或重新距离化方法，在保持零等值面固定的同时恢复符号距离性质。作为这些方法的替代方案，我们引入了一种改进的水平集平流方程，该方程能固有地保持界面处的梯度模长，即局部符号距离性质。数学上，这通过引入一个精心选择的源项来实现，该源项与局部界面面积生成率成正比。源项的引入使得问题变为非线性。然而，我们证明，通过显式时间离散化处理源项，每个时间步仅需求解线性方程即可。值得注意的是，在移动接触线情形下该方法无需额外调整即可正常工作。由于已知接触线涉及时会引发重新距离化问题，这构成一项重大优势。我们提供了该方法在二维和三维空间中基于简单一阶迎风格式的首次实现。