Essentially, I've come across a PDE for reaction-diffusion which describes the concentration of a substance at some given point in space:
$$\frac{\partial c_a}{\partial t} = \mu_a\nabla^2 c_a + R_a(c_a,c_b,c_c)$$
Where $\mu$ describes some diffusion coefficient, $c$ refers to the concentration of one of the 3 substances (at the same point in space), $a$, $b$ or $c$, and $R$ is a reaction equation to alter the concentration of substance $a$
I'd like to implement this reaction-diffusion system with discrete values, where $t$ only increases by 1 with each iteration, and all substances update simultaneously using values from the previous iteration, so I'm trying to understand how I can represent that in a clear way.
From what I've seen, there are multiple methods for solving PDEs with respect to space, but if I understand correctly, since this equation is concerned with only one point in space for all 3 substances, I should be concerned with solving this 'temporally'.
I understand that I can 'discretise' the first term using convolution with the right filter, and the reaction equation is quite straightforward, so my question is that am I right in thinking the following represents something closer to what I should be implementing?
$$c_a^{t+1} = c_a^t + \mu_a\nabla^2 c_a^{t} + R_a(c_a^t,c_b^t,c_c^t)$$
It seems quite straightforward, other than the convolution which I'm not sure how to represent, so I was wondering if I'm completely misunderstanding or trying to do the wrong thing