论文标题
高斯光束ANSATZ用于有限差波方程
Gaussian Beam ansatz for finite difference wave equations
论文作者
论文摘要
这项工作与用于波方程数的数值近似的高斯梁（GB）溶液的结构有关，该溶液通过有限的差异方案在空间中半污点。 GB是高频溶液，可以通过沿相应的哈密顿式的双段沿着双焦点工具在连续的和半分化水平上描述其传播。由于汉密尔顿人之间的高频差距，它们在连续和半差异的环境中的动力有所不同。特别是，数值高频解决方案可以表现出虚假的病理行为，例如空间缺乏传播，与连续波的经典时空传播特性相反。连续波和数值波之间的行为之间的差距也引入了重大的分析困难，因为经典的GB构建体不能立即将其外推到有限的差异设置，并且需要适当地定制以准确地检测离散媒体中的传播属性。本文我们的主要目的是为有限的差异方程介绍GB ANSATZ的一般且严格的结构，并通过准确的数值模拟来证实这种结构。
This work is concerned with the construction of Gaussian Beam (GB) solutions for the numerical approximation of wave equations, semi-discretized in space by finite difference schemes. GB are high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by microlocal tools along the bi-characteristics of the corresponding Hamiltonian. Their dynamics differ in the continuous and the semi-discrete setting, because of the high-frequency gap between the Hamiltonians. In particular, numerical high-frequency solutions can exhibit spurious pathological behaviors, such as lack of propagation in space, contrary to the classical space-time propagation properties of continuous waves. This gap between the behavior of continuous and numerical waves introduces also significant analytical difficulties, since classical GB constructions cannot be immediately extrapolated to the finite difference setting, and need to be properly tailored to accurately detect the propagation properties in discrete media. Our main objective in this paper is to present a general and rigorous construction of the GB ansatz for finite difference wave equations, and corroborate this construction through accurate numerical simulations.
