Δt/h2
framerate:fps
Diffusion on 512×512 square. Implicit scheme and relaxation
algorithm.
Diffusion equation
For the diffusion equation
∂t u = Δu ,
u|t=0 = exp(-(r -
ro )2/ a2),
the finite-difference scheme on square grid with the space step h
and time step dt is
ux,yt+1 =
ux,yt +
{[ux+1,y + ux-1,y + ux,y+1 + ux,y-1]
- 4ux,y } (Δt/h2).
Note that the time superscript is omitted in braces.
Explicit scheme
If we use superscript t in braces we get the simple explicit scheme
(the long term in the square brackets is symbolized as [ut])
ux,yt+1 =
ux,yt +
{[ut] - 4ux,yt} (Δt/h2) It is stable only for Δt/h2 < 1/4. So for small
h we need to use very small time step.
Implicit scheme and relaxation algorithm
Implicit scheme
ux,yt+1 =
ux,yt +
{[ut+1] - 4ux,yt+1} (Δt/h2) is stable for all Δt/h2. We rewrite it as
ux,yt+1(1 + 4Δt/h2)
= ux,yt + [ut+1] (Δt/h2) and solve iteratively starting with [ut]. Really the
application above makes only two iterations. Therefore it is not accurate
for large time steps.
Diffusion is not very impressive without convection...