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.

中文翻译：

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求解 Eikonal 方程的 Jet Marching 方法

我们开发了一系列用于求解特征方程的紧凑型高阶半拉格朗日标签设置方法。这些求解器推进 eikonal 的总 1-jet，并使用 Hermite 插值来近似每个半拉格朗日更新的 eikonal 和参数化特征。我们在任何维度的非结构化网格上描述求解器，并在二维规则网格上进行数值实验。我们的结果表明，这些求解器至少会产生二阶收敛，并且在特殊情况下，例如声速线性，eikonal 及其梯度的三阶收敛。我们还展示了如何使用基于单元格的插值来推进 eikonal 的第二部分。以这种方式计算的二阶导数信息通常是二阶准确的，适用于局部求解输运方程。这提供了一种推进来自 Helmholtz 方程的 WKB 近似的前因数的方法。这些求解器专为在声速缓慢变化的复杂环境中计算亥姆霍兹方程的高频近似值而设计，据我们所知，它们是第一个具有这些特性的求解器。我们提供了一个指向在线提供求解器的包的链接，从中可以轻松重现本文的结果。据我们所知，是第一个具有这些属性的求解器。我们提供了一个指向提供求解器的在线软件包的链接，从中可以轻松重现本文的结果。据我们所知，是第一个具有这些属性的求解器。我们提供了一个指向提供求解器的在线软件包的链接，从中可以轻松重现本文的结果。