# Explicit Euler stability for the Heat Equation (FDM)

Why the Explicit Euler scheme for the Heat Equation is stable only if $$k \leq h^2/2$$ ?

Here is the difference equation: $$$$\frac{U_j^{n+1}-U_{j}^n}{k} = \frac{1}{h^2}(U_{j+1}^n-2U_j^n+U_{j-1}^n)$$$$ with $$k$$ the mesh size in time and $$h$$ the mesh size in space

• To get some insight, rewrite your equation as $U^{n+1}_j = \frac{k}{h^2}U^n_{j+1}+ (1-2\frac{k}{h^2})U^n_{j} + \frac{k}{h^2}U^n_{j-1}$. The weights $(\frac{k}{h^2}, 1-2\frac{k}{h^2}, \frac{k}{h^2})$ can only be interpreted as a Markov chain transition probabilities iff $k \leq h^2/2$. Feb 27 '19 at 17:47
You can consult Seydel pages 99-106 for explicit FD or for a short summary this link. The idea is that you cannot choose $$k$$ and $$\frac{h^2}{2}$$ independently for stability.