I'm trying to solve the following system of PDEs using FDM.
My attempt is as follows:
$$ \begin{align*} c_{r,s}^{k+1} &= c_{r,s}^{k} + D \delta^2 \left( \mu_x + \mu_y \right) + f(c_{r,s}^{k}, m_{r,s}^k, n_{r,s}^k) \Delta t \\ n_{r,s}^{k+1} &= n_{r,s}^{k} + g(c_{r,s}^{k}, m_{r,s}^k, n_{r,s}^k) \Delta t\\ m_{r,s}^{k+1} &= m_{r,s}^{k} + h(c_{r,s}^{k}, m_{r,s}^k, n_{r,s}^k) \Delta t\\ \mu_x &= \Delta t/ \Delta x^2 \\ \mu_y &= \Delta t/ \Delta y^2 \\ \delta^2 &= c_{r+1,s}^{k} - 2 c_{r,s}^{k} + c_{r-1,s}^{k} \end{align*} $$
where $r$ and $s$ correspond to spatial dimensions and $k$ corresponds to time dimension.$f(.)$, $g(.)$ and $h(.)$ are some functions developed from the original equations given above. My confusion is as follows: the equations for $m$ and $n$ apparently don't have any spatial dimensions then how can I use their values in the equation to compute $c(x, y, t)$.