A finite periodic scalar-wave benchmark

Use the first circle factor of the fixed periodic three-torus from the earlier fixed-gauge data definition. This item is a separate nondimensionalized scalar model: both the spatial coordinate $x$ and time $t$ are dimensionless, the period is $2\pi$, and the model wave speed is $1$. Let $N\ge 3$ be an integer, $h:=2\pi \mathrm{/}N$, and $x_j:=jh(\mathrm{m}\mathrm{o}\mathrm{d}2\pi )$ for $j\in \mathbb{Z}\mathrm{/}N\mathbb{Z}$. A periodic grid function is an $N$-tuple $u=(u_j{)}_{j\in \mathbb{Z}\mathrm{/}N\mathbb{Z}}$ of real numbers. Define the centered periodic second-difference operator

where subscripts are reduced modulo $N$.

The algorithm takes as input such arrays $u^0,v^0$, a real time step $k>0$, and an integer $n_{\max }\ge 1$. It sets

then, for each integer $n=1,\dots ,n_{\max }-1$, computes

It outputs $(u^0,u^1,\dots ,u^{n_{\max }})$. All operations here are exact real arithmetic. No rounding-error model or claim about a floating-point implementation is part of the definition.

Fixed-gauge finite-slab data and solution topology

The recurrence is the centered finite-difference model for the scalar equation $u_{tt}=u_{xx}$. It is a one-dimensional diagnostic benchmark only; its stability properties do not establish convergence of a discretization of the Einstein equations. The initialization uses the second-order time expansion with the discrete spatial operator. The time step and mesh spacing are nondimensional, so their ratio has no unit conversion hidden in it.