The problem is let's say if I finite-difference one out of the many equations then through clank Nicholson, we separate the (t+1) time step left-hand side and (t) right side, then there are some other terms which are yet to be calculated.
In short - In coupled partial equation we are going to need the values of another dependent variable at the same time as we are calculating the value of the required variable.
If the above text won't clearly explain the problem then refer attached photo it will clearly speak of problem.
I wouldn't bother solving this or any other problem with MATLAB. Just write it up in C. It's much faster and compilers are free. You might find my book on Differential Equations helpful. It will be free on 8/18. The examples and source codes are always free. Crank-Nicholson is an interesting technique and has some advantage. There are other methods more easily implemented that can be quickly solved with modern computers. The computational advantage of C-N faded away with the mainframe. http://www.amazon.com/dp/B07DB9J2TK
What are your equations? What are your boundary conditions?
Many people use Crank-Nicolson for solving parabolic systems but I often wonder why. BDF2 has better stability properties with the same order of accuracy. Also, is there a reason why explicit time-stepping is being avoided as a solution method? An explicit scheme may need smaller timesteps and more of them, but it will be much easier to encode and the computation time per timestep is considerably smaller.
You have an under-determined system here as there are two variables, A and B, but only one equation. You need one more equation.
Once you have the two equations, you can use implicit or explicit scheme for solving them. With the implicit scheme, you need to solve a 2x2 block-matrix system in As and Bs.
D. Andrew S. Rees, I think that the popularity of the Crank-Nicolson scheme is due to its use in the literature, especially in the textbooks on numerical methods.
BDF2 scheme needs the solution from two previous time steps but for CN scheme requires only one previous solution. Some find it as a disadvantage of the BDF2 scheme. Not a big issue but still a valid argument. BDF2 scheme damps out all the high-frequency modes while CN scheme has no numerical damping; this means that, with BDF2, the timestep size needs to be smaller than that for the CN scheme to get accurate results. On the other hand, CN scheme can pollute the numerical results since it cannot damp out the high-frequency modes.
Both schemes have advantages and disadvantages. One has to study the results in detail to find the best scheme for the given PDE and the spatial discretisation method. For unstructured meshes and/or non-uniform discretisations, BDF2 tends to be a better choice.
You can easily decouple the discrete equations by computing every other dependent variable at the previous time level, t. Only the required dependent variable is computed at the new time level, t+1. This makes the problem easy to solve and even to parallelize.
See these papers:
1) https://www.emerald.com/insight/content/doi/10.1108/HFF-10-2019-0758/full/html
2)https://www.researchgate.net/profile/Chinedu_Nwaigwe2/publication/337812663_Computational_Analysis_of_Porous_Channel_Flow_with_Cross-Diffusion/links/5deb892fa6fdcc28370c964c/Computational-Analysis-of-Porous-Channel-Flow-with-Cross-Diffusion.pdf
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