I am solving PDEs of drift-difussion ion exchange (Nernst Plank equations) as shown in the attached picture (equation (1) and there are two more for B and D species). The dependent variables that I want temporally to solve are the concentration of A, B and D. I have discretized the generic equations using finite volume method for all three species and ended up with long equations ( eq(2) and eq(3) for A and B. As you notice, I have reduced some terms in order to simplify it and make it easier to read. Also, I have not included discretized formula of D since they are all the same). Anyway, I have six boundary conditons (two for each species) and initial conditions at t=0 for all. That is, all rhd terms are known at t=0 when solving for C at t=t+dt.
So my questions are:
1- Which ODE solver should I use? ODE45 and ODE15 solve IVP and bvp4c or bvp5c solve BVPs, but I have both conditions.
2- Each solver that I mentioned in the previous question tend to obtain the solution for just one spatial node, how can I make it work for the whole spatial domain?
3- The discretized ODEs I have are uncoupled since each term in rhs are known from perivous time step. From my poor knowledge, most of ODE solvers accept the coupled ones, so can I still use them? correct me if I am wrong.