M1 Mathematics, 2011-2012
MM26E Numerical Approximation of PDEs
Hervé Le Dret
The finite difference method for the heat equation
The forward Euler method in an unstable case
Below is an animation showing a computation of the heat equation solution using the explicit Euler method in a case when the discretization parameters do not satisfy the stability condition, but only by a tiny bit. Indeed, we have here $\frac{k}{h^2}=0.5154639>\frac12$.
The exact solution tends to $0$ exponentially fast. However, if we wait long enough, we see the numerical instability eventually develop. Round-up errors accumulate and are transferred to the unstable modes of the scheme, which ends up blowing up.
The animation shows the numerical solution up to time $T=1$. It then loops back at $t=0$. Computations performed with Scilab.
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