求解程函方程的喷流推进法
Jet Marching Methods for Solving the Eikonal Equation
摘要：
我们发展了一族求解程函方程的紧致高阶半拉格朗日标号设置方法。这些求解器对程函的总1-喷流进行处理，并使用Hermite插值来近似程函，并对每个半拉格朗日更新局部参数化特征。我们描述了求解器上的非结构化网格在任何维度，并进行了数值试验，在二维规则网格。我们的研究结果表明，这些求解器产生至少二阶收敛，并在特殊情况下，如线速度的声音，三阶收敛的程函和梯度。我们还展示了如何使用基于细胞的插值进行程函的第二部分。以这种方式计算的二阶导数信息通常是二阶精度的，适合于局部求解输运方程。这提供了一种方法来推进来自亥姆霍兹方程的WKB近似的前因子。这些解算器是专门为在具有缓慢变化的声速的复杂环境中计算亥姆霍兹方程的高频近似而设计的，据我们所知，它们是具有这些特性的第一个解算器。我们提供了一个提供求解器的在线软件包的链接，可以轻松地从中复制本文的结果。
英文摘要：
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.
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