Jet Marching Methods for Solving the Eikonal Equation

TL;DR

This paper introduces high-order semi-Lagrangian label-setting methods for solving the eikonal equation, achieving at least second-order convergence and extending to second derivatives, tailored for high-frequency Helmholtz equation approximations.

Contribution

The authors develop the first high-order semi-Lagrangian solvers for the eikonal equation that can handle unstructured meshes and compute second derivatives for Helmholtz applications.

Findings

Achieve at least second-order convergence in 2D

Third-order convergence in special cases like linear speed

First implementation of second-derivative marching for eikonal

Abstract

We develop a family of compact high-order semi-Lagrangian label-setting methods for solving the eikonal equation. These solvers march the total 1-jet of the eikonal, and use Hermite interpolation to approximate the eikonal and parametrize characteristics locally for each semi-Lagrangian update. We describe solvers on unstructured meshes in any dimension, and conduct numerical experiments on regular grids in two dimensions. Our results show that these solvers yield at least second-order convergence, and, in special cases such as a linear speed of sound, third-order of convergence for both the eikonal and its gradient. We additionally show how to march the second partials of the eikonal using cell-based interpolants. Second derivative information computed this way is frequently second-order accurate, suitable for locally solving the transport equation. This provides a means of marching the prefactor coming from the WKB approximation of the Helmholtz equation. These solvers are designed specifically for computing a high-frequency approximation of the Helmholtz equation in a complicated environment with a slowly varying speed of sound, and, to the best of our knowledge, are the first solvers with these properties. We provide a link to a package online providing the solvers, and from which the results of this paper can be reproduced easily.