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pythondifferential-equationspde

Alternating direction implicit method for finite difference solver of pde in Python

I am working on implementing the Alternating direction implicit method to solve FitzHugh–Nagumo reaction diffusion model. I have found a Python implementation example for it in a blog, but I think there is an error in the method - in the stencil presented here: enter image description here Shouldn't it be half time step size multiplying the reaction term f ?

Solution

  • Replacing the difference quotients by the differential quotients, one gets

    U_t = D/2 * U_xx + D/2 * U_yy + Δt*f

    in both instances, which is not the equation

    U_t = D * (U_xx + U_yy) + f

    that was the originally posed task.

    So the coefficients should be 1/(Δt/2) as it was at U_t, D/(Δp^2) at U_pp, p=x,y and 1 for f.

    It seems the formula is a mix-up of the one with difference quotients and the next stage where it gets multiplied by Δt/2.

    And in that next formula one does not need new constants as indeed α_p=σ_p, p=x,y and then you are right that the factor of f should be Δt/2.