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?