Crank-Nicolson scheme

The PDE is sampled at points $(x_i,t_{n+\half})$ (at the spatial mesh points, but in between two temporal mesh points). $$\frac{\partial}{\partial t} u(x_i, t_{n+\half}) = \dfc\frac{\partial^2}{\partial x^2}u(x_i, t_{n+\half})$$ for $i=1,\ldots,N_x-1$ and $n=0,\ldots, N_t-1$.

Centered differences in space and time: $$[D_t u = \dfc D_xD_x u]^{n+\half}_i$$