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## Main Question or Discussion Point

Hello, the work I am doing now requires numerical solution of several reaction-diffusion equations such as the standard advection diffusion equation (ADE) or IE Fisher's equation. So far I have been using the only method I am familiar with, the simplest one I believe: to express the time derivative as a simple Euler finite difference approximation, and to express the Laplacian using the following approximation:

[tex]\frac {d^2C} {dx^2} = \frac{C(x + 1) - 2C(x) + C(x - 1)}{dx^2}} [/tex]

where C is the density function (the variable diffusing), which is here a 1d matrix and dx is the step size. I approximated dC/dx with a backward difference approximation, and I approximated [itex]\frac {d^2C} {dx^2}[/itex] with a forward difference approximation, then combined these to get the approximation above.

These techniques have worked for what I'm doing fairly well until now, but I would like to try using more accurate numerical methods. I am wondering what a simple, but higher order or more accurate method would be. Can anyone give me a name of a method? Can anyone recommend any good numerical methods books that I might be able to get through my university library or an interlibrary loan?

Also for a more accurate method would I have to keep track for more than just the state of the 1d matrix at the time before? IE would I have to keep track of the last 2 iterations, etc.

Thank you.

[tex]\frac {d^2C} {dx^2} = \frac{C(x + 1) - 2C(x) + C(x - 1)}{dx^2}} [/tex]

where C is the density function (the variable diffusing), which is here a 1d matrix and dx is the step size. I approximated dC/dx with a backward difference approximation, and I approximated [itex]\frac {d^2C} {dx^2}[/itex] with a forward difference approximation, then combined these to get the approximation above.

These techniques have worked for what I'm doing fairly well until now, but I would like to try using more accurate numerical methods. I am wondering what a simple, but higher order or more accurate method would be. Can anyone give me a name of a method? Can anyone recommend any good numerical methods books that I might be able to get through my university library or an interlibrary loan?

Also for a more accurate method would I have to keep track for more than just the state of the 1d matrix at the time before? IE would I have to keep track of the last 2 iterations, etc.

Thank you.