I am trying to solve the 2-D heat equation $\frac{\partial u(x,y,t)}{\partial t}=\nabla^2 u(x,y,t)$ by approximating the spatial differential operator i.e $\nabla^2 u$. The approximation yields a differentiation matrix too large to post here. I solved the system of ODEs by decomposing the matrix into its eigensystem. A similar approach to what has been done here.
The initial conditions are shown to be satisfied below.
The problem I encounter is that when I step the solution in time the boundary values 'blow up' and 'cold spot' migrates to the center of the domain for all time.
I simply expected the temperature to smooth out and form a level surface. Does this look like a possible solution or is it clear there must be a large numerical error evolved. Would one ever expect this sought of surface to satisfy the heat equation, I get the feeling its just not 'right'.
The initial conditions I used where generated in Mathematica and are about 441 equations, for example below are the first 5.
- u[0., 0., 0.] == 0.01
- u[0., 0.1, 0.] == 0.02
- u[0., 0.2, 0.] == 0.03
- u[0., 0.3, 0.] == 0.04
- u[0., 0.4, 0.] == 0.05
The answer was no, one should expect smoothing along the boundaries. This is not an intuitive solution to the heat equation for reasons better outlined in a similar question