(adsbygoogle = window.adsbygoogle || []).push({}); To the moderator: please move this to the section on differential equations if you think it would be better there.

I'm looking at a reaction-diffusion equation of the form

[tex]\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(u),[/tex]

where, e.g., [tex]f(u) = u(1-u)[/tex] for the Fisher-Kolmogorov eqation, and I'm assuming the spatial domain is the unit interval with Neumann boundary conditions:

[tex]\frac{\partial u}{\partial x}(0, t) = 0, \; \frac{\partial u}{\partial x}(1, t) = 0, \; t\geq 0.[/tex]

To keep things totally simple, I'm using the backward Euler method. When the original equation is just the heat equation, i.e., [tex]f = 0[/tex], this method just involves solving a simple matrix equation,

[tex]BU^{n+1} = U^n,[/tex]

for evolving the solution from time step n to n+1. The matrix of course depends on the boundary conditions. The nonlinear version of Euler's method involves a nonlinear multi-variable equation:

[tex]BU^{n+1} + F(U^{n+1}) = U^n,[/tex]

which can be solved for each time step using Newton's method.

Where I'm getting confused is: exactly how do I incorporate the boundary conditions when deriving the system of nonlinear equations? My reasoning tells me that they should be the same as for the linear equation, i.e., the matrices B are the same for both the linear and nonlinear systems.

My solver works perfectly for the heat equation with Neumann conditions, but gives only garbage for the Fisher-Kolmogorov equation after the first time step (if it converges at all). I'm convinced that the problem is not a bug in programming Newton's method. Anyway I'm using a library for that, and I've checked the functions I've supplied a thousand times already. Anyone here have experience with nonlinear parabolic equations? They're not the kind of thing that's typically covered in basic numerical analysis texts.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Nonlinear parabolic equations: finite difference method

**Physics Forums | Science Articles, Homework Help, Discussion**