Uniform periodic, infinite, and Dirichlet grid conventions

Use $\mathbb{N}$ for the natural-number index set (including zero) (The natural numbers $\mathbb{N}$ (von Neumann)), $\mathbb{R}$ for the real scalar field (The real numbers), and $\mathbb{C}$ with its real embedding and modulus (The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$, Real and imaginary parts, complex conjugation, and modulus).

Let $N\ge 2$, $h>0$, $k>0$, and $t_n=nk$. For the finite periodic grid put $J_N=\{0,\dots ,N-1\}$ and reduce every shifted index modulo $N$. A periodic grid function is a map $v:J_N\to \mathbb{C}$, with weighted norm $\mathrm{\parallel }v{\mathrm{\parallel }}_{2,h}={\left(h{\sum }_{j=0}^{N-1}\mathrm{\mid }{v}_j{\mathrm{\mid }}^2\right)}^{1\mathrm{/}2}\mathrm{.}$ The infinite uniform grid has indices $j\in \mathbb{Z}$ and positions $x_j=jh$. On the periodic and infinite grids define $(D_{+}v{)}_j=\frac{v_{j+1}-v_j}{h},(D_{-}v{)}_j=\frac{v_j-v_{j-1}}{h},(D_{0}v{)}_j=\frac{v_{j+1}-v_{j-1}}{2h},({\mathrm{\Delta }}_{h}v{)}_j=\frac{v_{j+1}-2v_j+v_{j-1}}{h^2}\mathrm{.}$ Use the inner product linear in its first entry, $⟨p,q{⟩}_{2,h}=h{\sum }_{j=0}^{N-1}{p}_j\overline{q_j},⟨D_{+}p,D_{+}q{⟩}_{e,h}=h{\sum }_{j=0}^{N-1}(D_{+}p{)}_j\overline{(D_{+}q{)}_j},$ where the indices in $D_{+}$ are reduced periodically, and set $\mathrm{\parallel }{D}_{+}p{\mathrm{\parallel }}_{e,h}^2=⟨D_{+}p,D_{+}p{⟩}_{e,h}$. The same definitions apply to real-valued grid functions.

For a finite interval of length $L=Nh$, use physical nodes $x_j=jh$ with indices $j=0,\dots ,N$. On its edges $j=0,\dots ,N-1$ define $(D_{+}v{)}_j=(v_{j+1}-v_j)\mathrm{/}h$, and on interior nodes $j=1,\dots ,N-1$ use $({\mathrm{\Delta }}_{h}v{)}_j=(v_{j+1}-2v_j+v_{j-1})\mathrm{/}{h}^2$. At the right endpoint define $(D_{-}v{)}_N=(v_N-v_{N-1})\mathrm{/}h$. For interior-node arrays $r,s$ indexed by $j=1,\dots ,N-1$ and edge arrays $p,q$ indexed by $j=0,\dots ,N-1$, define $⟨r,s{⟩}_h=h{\sum }_{j=1}^{N-1}{r}_j\overline{s_j},\mathrm{\parallel }r{\mathrm{\parallel }}_h^2=⟨r,r{⟩}_h,⟨p,q{⟩}_{e,h}=h{\sum }_{j=0}^{N-1}{p}_j\overline{q_j}\mathrm{.}$ Set $\mathrm{\parallel }p{\mathrm{\parallel }}_{e,h}^2=⟨p,p{⟩}_{e,h}$. For nodal grid functions, interior pairings and norms mean restriction to the interior nodes. Thus $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_h$ is a norm on the zero-endpoint subspace, but only a seminorm on the full nodal space. Dirichlet grid functions have prescribed endpoint values; when they are zero, the interior norm and edge inner product above are the ones used for the wave energy.

All statements below distinguish these periodic, infinite, and finite-interval index sets. Periodic Fourier calculations are not boundary estimates for the finite-interval grid.