I'm dealing with a particular reaction-diffusion equation having the form

$$ c_t = \alpha \nabla^2 c + F(c,x,t). \tag{1}$$

where $F$ is nonlinear. I would like to solve (1) with a finite-difference scheme. Specifically, I would like to use a Crank-Nicolson scheme to progress the solution forward in time. Letting $C_{i,j}^n \approx c(x_i,y_j,t_n)$ be an approximation to $c$ at the grid point $(x_i,y_j,t_n)$, (1) can be discretized as

$$ \frac{C_{ij}^{n+1} - C_{ij}^n}{\delta t} = \frac{1}{2} (\nabla_h^2C_{ij}^{n} + \nabla_h^2C_{ij}^{n+1}) + \frac{1}{2}(F_{ij}^n + F_{ij}^{n+1}), \tag{2}$$

where $\nabla_h^2$ is the discretized Laplacian. For reasons I won't get into here, it's impossible to use Picard iteration/Newton's method to linearize (2). My question, then, is the following - if I were to modify the Crank-Nicolson scheme in (2), using $F_{ij}^n$ instead of $(1/2)(F_{ij}^n + F_{ij}^{n+1})$, could I say anything about the convergence of the resulting scheme?

  • 1
    $\begingroup$ Splitting methods are frequently used, applying different method steps to the linear part and the non-linear part. Your modification reduces the order to that of the Euler method, using the trapezoidal method in the stiff non-linear part might provide extra stability that is not captured by the order alone. It should be possible to increase the order for the non-linear part back to 2 using Heun/explicit trapezium. $\endgroup$ Dec 13, 2023 at 10:54


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