Numerical solution of Fisher's equation

[mu00]

Homework Statement

Solve Fisher's equation
u_t = u_{xx} + u(1 - u)
numerically, with the initial condition (step function)
u(x,0) = heaviside(-x)

Homework Equations

One can assume a traveling wave solution:
u(x,t) = u(\xi)
where \xi = x-vt
such that
u_{\xi \xi} = -vu_{\xi} - u(1 - u)
which is a second order nonlinear ode (right?)

The Attempt at a Solution

I have tried to implement the split-step (pseudospectral) method, but I ran into trouble when I tried to deal with the nonlinear term u^2. I tried the (implicit) Crank-Nicolson method - had difficulty there too. I am now attempting the (explicit) Newton-Kantorovich method - not making much progress.

The traveling wave solution should be a helpful simplification, though I'm not sure what the next step is: maybe a substitution of the form w = u_{\xi}, to reduce to a first order nonlinear ode (?).
Would a straightforward explicit finite differences method work or will I need something more sophisticated?

Any help will be fantastic!

 

[takbq2]

I'm interested in this too... despite it being posted a long time ago.