Discretised wave equations

Can anyone please explain this to me? I can’t really figure out what the symbols mean.

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The wave equation is usually expressed in the form

$latex displaystyle partial_{tt} u – Delta u = 0&fg=000000$

where $latex {u colon {bf R} times {bf R}^d rightarrow {bf C}}&fg=000000$ is a function of both time $latex {t in {bf R}}&fg=000000$ and space $latex {x in {bf R}^d}&fg=000000$, with $latex {Delta}&fg=000000$ being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain $latex {{bf R}^d}&fg=000000$ with some other manifold and replacing the Laplacian $latex {Delta}&fg=000000$ with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on $latex {{bf R}^d}&fg=000000$. We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy


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