M1 Mathematics, 2011-2012
MM26E Numerical Approximation of PDEs
Hervé Le Dret
The finite difference method for the heat equation
The backward Euler method
Below is an animation showing a computation of the heat equation solution using the implicit Euler method, thus unconditionally stable. Here we have $\frac{k}{h^2}=200$.
The initial data does not satisfy the homogeneous Dirichlet condition (it is equal to $1$ on $[0,1]$). The exact solution is thus not regular at $t=0$. This does not prevent the method from converging quite well.
The animation is cut well before $t=1$, since nothing spectacular happens, then loops back at $t=0$. Computations performed with Scilab.
|