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.

翻译：我们提出了一种新的嵌入方法，用于求解曲面上的标量双曲守恒律。该方法沿用典型水平集方法及某些嵌入方法，通过符号距离函数隐式表示界面。不同于直接在曲面上显式求解方程，我们在界面附近的一个小邻域内引入了一个修正的偏微分方程。该嵌入方程基于一种推进算子建立，该算子能够将任何切向通量向量从曲面扩展至相邻的水平面。该算子易于计算，仅涉及水平集函数及其对应的Hessian矩阵。所得解在界面的法向上保持恒定。为展示我们方法的精度与有效性，我们给出了若干二维和三维算例。