We analyze the propagation properties of the numerical versions of one- and two-dimensional wave equations, semi-discretized in space by finite difference schemes. We focus on high-frequency solutions whose propagation can be described, both at the continuous and at the semi-discrete levels, by micro-local tools. We consider uniform and non-uniform numerical grids as well as constant and variable coefficients. The energy of continuous and semi-discrete high-frequency solutions propagates along bi-characteristic rays, but their dynamics are different in the continuous and the semi-discrete setting, because of the nature of the corresponding Hamiltonians. One of the main objectives of this paper is to illustrate through accurate numerical simulations that, in agreement with micro-local theory, numerical high-frequency solutions can bend in an unexpected manner, as a result of the accumulation of the local effects introduced by the heterogeneity of the numerical grid. These effects are enhanced in the multi-dimensional case where the interaction and combination of such behaviors in the various space directions may produce, for instance, the rodeo effect, i.e., waves that are trapped by the numerical grid in closed loops, without ever getting to the exterior boundary. Our analysis allows to explain all such pathological behaviors. Moreover, the discussion in this paper also contributes to the existing theory about the necessity of filtering high-frequency numerical components when dealing with control and inversion problems for waves, which is based very much on the theory of rays and, in particular, on the fact that they can be observed when reaching the exterior boundary of the domain, a key property that can be lost through numerical discretization.

我们分析了有限差分格式在空间上半离散的一维和二维波动方程数值版本的传播特性。我们专注于高频解决方案，其传播可以通过微局部工具在连续和半离散水平上进行描述。我们考虑统一和非统一的数值网格以及常数和可变系数。连续和半离散高频解的能量沿双特征射线传播，但是由于相应的哈密顿量的性质，它们在连续和半离散设置中的动力学是不同的。本文的主要目的之一是通过与微观局部理论一致的精确数值模拟来说明，由于数值网格的不均匀性导致的局部效应的累积，数值高频解可能会以意想不到的方式弯曲。在多维情况下，这些效果会增强，在多维情况下，此类行为在各个空间方向上的交互作用和组合可能会产生例如牛仔竞技表演，即被数字网格闭环包围的波，而从未到达外部边界。我们的分析可以解释所有此类病理行为。此外，本文的讨论还为现有的关于在处理波的控制和反演问题时过滤高频数值分量的必要性理论做出了贡献，该理论很大程度上基于射线理论，尤其是基于事实上，到达域的外部边界时可以观察到它们，这是一个关键属性，可以通过数值离散化来丢失。