A Simple Embedding Method for Scalar Hyperbolic Conservation Laws on Implicit Surfaces

TL;DR

This paper introduces a new embedding method for solving scalar hyperbolic conservation laws on surfaces using level set functions and a modified PDE in a neighborhood of the interface, demonstrated through 2D and 3D examples.

Contribution

The paper presents a novel embedding approach that simplifies solving conservation laws on surfaces by extending flux vectors via a push-forward operator.

Findings

Method accurately solves conservation laws on surfaces.

The approach is computationally efficient and easy to implement.

Validated with multiple 2D and 3D examples.

Abstract

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.