I'm trying to solve coupled PDE by Crank-Nicolson (CN) and Newton-Raphson method with MATLAB. I have used CN method but not for coupled problem. Please if someone could help let me know to add more details about the equation.
Thanks in advance
Detail of coupled equation:
I’m going to solve Poisson-Nernst-Planck system of equations. The Nernst-Planck equation is $$\frac{\partial n_i}{\partial t}=A_i\frac{\partial^2 n_i}{\partial x^2}+B_i\frac{\partial n_i}{\partial x}\frac{\partial \phi}{\partial x}+B_i(n_i\frac{\partial^2 \phi}{\partial x^2}),$$ where $n_i$ is the concentration of specific ion, $A_i$ and $B_i$ is the coefficient of equation for specific ion, $\phi$ is the electrical potential. The Poisson equation that couples by the previous equation is $$\epsilon \frac{\partial^2 \phi}{\partial x^2}=F\sum(n_i*z_i),$$ where $\epsilon$ is the permittivity of medium, $F$ is the Faraday constant, $n_i$ is the concentration of specific ion and $z_i$ is the ionic charge. So how could I implement Poisson equation into Nernst-Planck and after that implement Newton-Raphson method for linearize it. Or do we need Newton-Raphson method to solve this equation? or we could solve it just by CN method. Please let me know if anybody needs more details or if something is unclear.