隐式曲面上标量双曲守恒律的一种简单嵌入方法
A Simple Embedding Method for Scalar Hyperbolic Conservation Laws on Implicit Surfaces
AI总结：
本文提出一种基于符号距离函数与 push-forward 算子的简单嵌入方法，在界面邻域求解修正偏微分方程，以准确高效地模拟隐式曲面上的标量双曲守恒律。
摘要：
我们开发了一种新的嵌入方法，用于求解曲面上的标量双曲守恒律。该方法遵循典型的 level set 方法和一些嵌入方法，通过符号距离函数隐式表示界面。我们并非在曲面上显式求解方程，而是在界面的一个小邻域内引入一个修正的偏微分方程。该嵌入方程基于一个 push-forward 算子构建，该算子能够将任意切向通量向量从曲面扩展到相邻的 level 曲面。此算子易于计算，只涉及 level set 函数及其相应的 Hessian。所得解在界面的法向方向上为常数。为证明我们方法的精度和有效性，我们给出了若干二维和三维算例。
英文摘要：
We have developed a new embedding method for solving scalar hyperbolic conservation laws on surfaces. The approach represents the interface implicitly by a signed distance function following the typical level set method and some embedding methods. Instead of solving the equation explicitly on the surface, we introduce a modified partial differential equation in a small neighborhood of the interface. This embedding equation is developed based on a push-forward operator that can extend any tangential flux vectors from the surface to a neighboring level surface. This operator is easy to compute and involves only the level set function and the corresponding Hessian. The resulting solution is constant in the normal direction of the interface. To demonstrate the accuracy and effectiveness of our method, we provide some two- and three-dimensional examples.
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