The algorithm only needs to access the three most recent time levels, so we need only three arrays for $u_i^{n+1}$, $u_i^n$, and $u_i^{n-1}$, $i=0,\ldots,N_x$. Storing all the solutions in a two-dimensional array of size $(N_x+1)\times (N_t+1)$ would be possible in this simple one-dimensional PDE problem, but not in large 2D problems and not even in small 3D problems.