Hi I have made a 3D alternating direction explicit scheme for solving the diffusion equation, which will eventually replace a FTCS scheme in model of bubble dynamics in tissue. I have been testing it and have realized that despite the unconditional stability I get oscillations in the system if the Courant number is greater than ~4. Is there a way round this other than reducing the Courant number?
Here are some more details: The scheme I am using alternating direction explicit ADE is an explicit method which does two sweeps (one upwind and one downwind). In 1D this looks like:
C1_new[i]-C_old[i] = (dt/h^2)(C_old[i+1]-C_old[i]-C1_new[i]+C1_new[i-1])
C2_new[i]-C_old[i] = (dt/h^2)(C_old[i-1]-C_old[i]-C2_new[i]+C2_new[i+1])
C_new[i] = C1_new[i]+C2_new[i]/2
Full details can be found at http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1552926
However, when I implement a 3D version and play around with time and space steps I see oscillations occurring from (dt/h^2) values of about 4.5 upwards.
This is a plot of the pt (5,5,5) from my $10\times10\times10$ grid for various Courant numbers. As you can see, there are oscillations for the higher numbers.