The model problem
We consider the following partial differential equation with initial and boundary conditions
|Initial Condition for the model problem.|
All eigenvalues positive
If we set lambda_1 = 1, lambda_2 = 0.1, and take 50 grid points in every space direction and a time step of 1/10000, then we obtain at t = 1
Note the scale: The Dirichlet condition V=0 at all boundaries draws the solution towards zero.
Second eigenvalue negative
For lambda_1 = 1, lambda_2 = - 0.1, we obtain for t=0.02, 0.04, 0.06:
The solution explodes. The scale on the third plot is 10^19.
Negative eigenvalue closer to zero
Maybe the second eigenvalue was "too negative"? What happens for lambda_2 = - 0.01?
The reason for these oscillations does not lie in the explicit scheme but in the ill-posedness of the equation. In the Traunsee example, the instability of the backwards heat equation was analysed in more detail.