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

  • $\begingroup$ please have a look at the links in my answer. If you are satisfied, please mark the answer as the solution. $\endgroup$ – FunnyBuzer Feb 26 '19 at 16:46
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    $\begingroup$ 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 $k \leq h^2/2$. $\endgroup$ – Antoine Conze 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.


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