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I want to solve the following system of nonlinear reaction-diffusion equations (Schnakenberg Turing) using FEM methods (such as deal.ii):

$$ \partial_{t} u = \Delta u + \gamma\left(a-u+u²v\right)$$ $$ \partial_{t} v = d\Delta v + \gamma\left(b-u²v\right)$$

where d, $\gamma$, a, b being constants.

Probably I need to first apply a time integration scheme (such as Crank-Nicholson, implicit Runge-Kutta), and then FE space discretization.

Question: How should I choose and perform the time integration for this nonlinear system? Any hints for very similar examples or specific literature?

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As a first pass I would suggest a splitting scheme. Do an implicit timestepping scheme (e.g. CN) for the diffusion part for both u and v equations. Then do an explicit step for the nonlinear terms (e.g. RK).

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