(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.

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# Nonlinear parabolic equations: finite difference method

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