Solving the 3D Diffusion Equation with Fourier Spectral Techniques

[johnnyTransform]

Hi guys,

I've distilled the 3D Diffusion Equation into the following PDE using Fourier spectral techniques:

∂C(m,n,p,t)/∂t + k(p^2+m^2+n^2)C(m,n,p,t)=0,

where C is the Fourier coefficient of the 3D Fourier transform, {m,n,p} are the spatial frequencies, and t is time. I've tried using a simple explicit scheme:

C(m,n,p)^v+1=1/(1+k*deltaT*(p^2+m^2+n^2)*C(m,n,p)^v

where v+1 is the leading time step, and v is the current time step. However, it seems to simply decay to zero over time. Any suggestions as to how I could treat it?

 

[Simon Bridge]

Since you only care about the t dependence for the DE, this is form $\frac{dy}{dt} + ay = 0$ which can be solved exactly.

 

[johnnyTransform]

Thanks!