I am building a model which, among other things, needs to solve the following PDE multiple times:
$$\frac{\partial \phi}{\partial t} = D_{o} \nabla^2 \phi + s_o - k_o$$
Where $s_o$ and $k_o$ are respectively source and sink grids.
Note that because the equation above represents diffusion of oxygen, which is significantly faster than the biological processes that the model simulates, I could also use the adiabatic solution:
$$D_{o} \nabla^2 \phi + s_o - k_o = 0$$
The equation is to be solved in three dimensions.
I am currently using FiPy (https://www.ctcms.nist.gov/fipy) that uses the Finite Volume Method, which takes about 10 seconds for a 50x50x50 grid with $dx = dy = dz = 1$.
I am not an expert in calculus, but was wondering if the above equation had a direct/one-step solution or if I was limited to numerical methods?
Boundary conditions are no-flux.