In this paper, a new numerical computational frame is presented for solving moving interface problems modeled by parabolic PDEs on static and evolving surfaces. The surface PDEs can have Dirac delta source distributions and discontinuous coefficients. One application is for thermally driven moving interfaces on surfaces such as Stefan problems and dendritic solidification phenomena on solid surfaces. One novelty of the new method is the local tangential lifting method to construct discrete delta functions on surfaces. The idea of the local tangential lifting method is to transform a local surface problem to a local two dimensional one on the tangent planes of surfaces at some selected surface nodes. Moreover, a surface version of the front tracking method is developed to track moving interfaces on surfaces. Strategies have been developed for computing geodesic curvatures of interfaces on surfaces. Various numerical examples are presented to demonstrate the accuracy of the new methods. It is also interesting to see the comparison of the dendritic solidification processes in two dimensional spaces and on surfaces.

在本文中，提出了一种新的数值计算框架，用于解决由抛物线形偏微分方程在静态和不断变化的表面上建模的运动界面问题。表面PDE可以具有Diracδ源分布和不连续系数。一种应用是用于表面上的热驱动运动界面，例如Stefan问题和固体表面上的树枝状凝固现象。新方法的一种新颖性是局部切向提升方法，可在表面上构造离散的增量函数。局部切向提升方法的思想是将局部表面问题转换为在某些选定表面节点处的表面的切平面上的局部二维问题。此外，开发了前跟踪方法的表面版本来跟踪表面上的移动界面。已经开发出用于计算表面上的界面的测地曲率的策略。给出了各种数值示例，以证明新方法的准确性。同样有趣的是，在二维空间和表面上比较了树枝状凝固过程。