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

• please have a look at the links in my answer. If you are satisfied, please mark the answer as the solution. – FunnyBuzer Feb 26 at 16:46
• 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 $k \leq h^2/2$. – Antoine Conze Feb 27 at 17:47

## 1 Answer

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.