Centered leapfrog wave recurrence with Dirichlet and periodic rows

Use the uniform-grid indices, physical nodes, and spacings of Uniform periodic, infinite, and Dirichlet grid conventions. Fix a wave speed $c>0$, an interval length $L=Nh$ with $N\ge 2$, and a time step $k>0$. Write $\nu =ck\mathrm{/}h$. The target model is the real scalar wave equation $u_{tt}=c^2{u}_{xx}$, and grid states are real-valued. Prescribe a real initial displacement row $U^0$ and discrete initial-velocity row $V^0$, and set $U^1=U^0+k{V}^0$. For smooth Dirichlet data $u_0,u_1:[0,L]\to \mathbb{R}$ with $u_0(0)=u_0(L)=u_1(0)=u_1(L)=0$, the corresponding sample rows are $U_j^0=u_0(x_j)$ and $V_j^0=u_1(x_j)$; for smooth periodic data, take real $L$-periodic $u_0,u_1$ and sample them at $x_j$. In both cases the discrete initial velocity equals the sampled $u_1$. The centered leapfrog recurrence is $U_j^{n+1}-2U_j^n+U_j^{n-1}={\nu }^2(U_{j+1}^n-2U_j^n+U_{j-1}^n),n\ge 1.$ For the finite Dirichlet interval, require $U_0^0=U_N^0=V_0^0=V_N^0=0$ and impose the recurrence for $j=1,\dots ,N-1$, with $U_0^n=U_N^n=0$ for every $n\ge 0$. For the periodic branch, use $j=0,\dots ,N-1$, reduce indices modulo $N$, and impose the recurrence at every node. Fourier analysis of this real periodic update uses its complex-linear extension to the complexified grid space. No convergence order for a particular initialization formula is asserted here.