FTCS scheme

Remarks

Its local truncation error is $O(\Delta t + \Delta x^2)$ for smooth solutions. Von Neumann analysis gives the amplification factor $g(\xi) = 1 - 4 r \sin^2(\xi/2)$, so the scheme is stable if and only if $r \le 1/2$. For $r \le 1/2$ the new value $u_j^{n+1} = (1 - 2r) u_j^n + r u_{j+1}^n + r u_{j-1}^n$ is a convex combination of old values, which gives a discrete maximum principle. Example: with $r = 1/2$ the scheme reads $u_j^{n+1} = (u_{j+1}^n + u_{j-1}^n)/2$. Starting from $u^0_0 = 1$ and $u^0_j = 0$ for $j \ne 0$, one step gives $u^1_{\pm 1} = 1/2$ and all other values $0$; the total heat $\sum_j u_j^n$ stays $1$. Non-example: the Crank–Nicolson scheme also uses centred differences in space, but it evaluates them at the new time level as well. So it is implicit and is not the FTCS scheme. The same forward-time, centred-space construction applied to the advection equation $u_t + a u_x = 0$ with $a \ne 0$ is unstable for every ratio $\Delta t / \Delta x$.