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.

这项工作涉及构建波动方程数值近似的高斯梁 (GB) 解，通过有限差分格式在空间中进行半离散化。GB 是高频解，其传播可以通过微局域工具沿着相应哈密顿量的双特征在连续和半离散水平上进行描述。由于哈密顿量之间的高频间隙，它们的动力学在连续和半离散设置中有所不同。特别是，数值高频解可能表现出虚假的病理行为，例如缺乏空间传播，这与连续波的经典时空传播特性相反。连续波和数值波的行为之间的差距也带来了显着的分析困难，因为经典的 GB 结构不能立即外推到有限差分设置，并且需要适当调整以准确检测离散介质中的传播特性。我们本文的主要目标是提出有限差分波动方程的 GB ansatz 的通用且严格的构造，并通过精确的数值模拟证实该构造。