I have a system of coupled nonlinear PDEs that I cannot figure out how to solve in a smart way using FDM, so I was hoping someone here might have a clue. The equations go as:

\begin{align*} \frac{1}{n_\alpha}\frac{\partial}{\partial t} \left[n_\alpha-A_\alpha\left(\nabla^2\phi+\nabla^2 p_\alpha\right) \right] &=\text{RHS}_{n_\alpha} \\ \frac{\partial}{\partial t} \left[\ln(p_\alpha)- B_\alpha\left(\nabla^2\phi+\nabla^2 p_\alpha\right) \right] &=\text{RHS}_{p_\alpha}\\ \frac{\partial}{\partial t} \left[\sum_\alpha C_\alpha\left(\nabla^2\phi+\nabla^2 p_\alpha\right)\right] &= \text{RHS}_\text{vort}\\ \end{align*} Where $A_\alpha, B_\alpha, C_\alpha$ are constants and $\alpha$ indicates the "species". I.e. it could range from $1$ to $n$ and so there would in the end be $2n+1$ equations that need to solved.

I suppose one would start with time stepping the equations, so one "only" needs to solve what seems to be coupled with nonlinear Helmholtz-type equations. But then what?

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    $\begingroup$ There's not enough information here to give a useful answer. Several important things are undefined. At least the dependencies need to be specified. $\endgroup$ – David Ketcheson May 19 '18 at 11:11

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