I was recently looking at this Wikipedia page, and specifically the section explicit method for the heat equation. To summarise, it says we can use the numerical scheme below.
We have the differential equation $$U_t=U_{xx}$$ We want to numerically approximate the points at $(x_j,t_n)$ by $u_j^n$. By expressing the derivatives as $$U_t\approx\frac{u_j^{n+1}-u_j^n}{k}\text{ and }U_{xx}\approx\frac{u_{j+1}^n-2u_j^n+u_{j-1}^n}{h^2},$$ we can rearrange to get $$u_j^{n+1}=(1-2r)u_j^n+r u_{j-1}^n+r u_{j+1}^n$$ where $r=k/h^2$
What I don't understand is why this is stable for $r\leq 1/2$ and why it would be unstable if $r>1/2$ - what causes this instability? Is it something to do with the $1-2r$ term, which we need to keep $\leq0$? If so, why?
As Eff pointed out, Von Neumann analysis is the way to go. Suppose $u_j^n = z^n e^{i j \xi}$. Plugging this into the equation $$ z = (1-2r) + r( e^{-i \xi} + e^{i \xi}) = (1-2r) + 2r \cos \xi. $$ The stability is only if $|z| < 1$ which implies $$ -1 < r (\cos \xi - 1) < 0. $$ Note here, that $\cos \xi - 1 \le 0$ and so $$ \frac{1}{1 - \cos \xi} > r > 0. $$ Finally, we want stability for all $\xi$ and so we take the minimum for the LHS which gives $$ \frac{1}{2} > r > 0. $$