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Can you use the Crank-Nicolson method to get a numerical approximation to the fisher-kolmogorov equation? If not what would be the easiest way to model the equation using matlab?

Thanks and sorry its so basic but i'm new to these sorts of numerical approximations

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2 Answers 2

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Crank-Nicolson will work fine, though a fully implicit scheme may actually perform a bit better especially if you have discontinuous initial conditions. Since Crank-Nicolson is effectively half explicit, discontinuities in the initial conditions $u(x,0)$ propagate into a lot of noise echoing around your solution grid.

Make sure you choose $\Delta t$ sufficiently small that your propagation matrix remains diagonally dominant. That is to say, choose it small in comparison to the initial size $|u(x,0)|$.

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Another approach especially in higher dimensions would be to use a Fourier - Spectral method for spatial discretization followed by a fourth order Runge - Kutta algorithm to the algebraic differential system which is in "Fourier Space". The Inverse Fast Fourier Transform can then be used to recover the solution in the original domain. This technique is useful especially with coupled reaction diffusion equations.

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