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.

翻译：我们开发了一套用于解决电离方程式的精密高阶半拉格朗吉亚标签设定方法。 这些解答器在电离子方程式的总和1喷射中行进总和, 并使用Hermite内插法近似每个半拉格朗吉亚更新的当地电离线和半拉格朗吉亚特性。 我们描述在任何维度上未结构的网状的解答器的解答器, 在常规网格上进行两个维度的数值实验。 我们的结果表明, 这些解答器至少产生二阶趋同, 在音速等特殊情况下, eiikonal 及其梯度的第三级趋同。 我们还展示了如何使用基于细胞的内插器来接近电离子的第二个部分。 计算精度的第二个衍生信息通常为二阶准确, 适合本地解运方程式。 这提供了一种从Helmholtz方程式的WKB近似速度, 这些解答器专门设计用于计算Helmholtz方程式的高频近近近, 和速度, 等解器的精度方方程式在一种复杂环境中, 提供我们最复杂的软件, 的精密的解到这些解到网络的解器的精度。