Scheme to solve Maxwell's equations coupled to charge continuity equation?

Seems like most better known schemes don't handle this particular situation, so could someone suggest a good finite difference scheme to solve the following coupled equations (Maxwell+continuity equation for charge density): \begin{align} &\frac{\partial \vec{B}}{\partial t} = -\nabla \times\vec{E} \\&\frac{\partial \vec{E}}{\partial t} = \nabla \times\vec{B}-\rho\vec{v} \\&\frac{\partial \rho}{\partial t}+\nabla \cdot(\rho\vec{v})=Q \\& \vec{v} = \vec{f}(\vec{E},\vec{B}) \end{align} Any help you could give would be appreciated.

• In standard Yee's scheme for isotropic material, in the second equation for $\nabla \times \vec{B}$ you have a known function $\vec{J}$. Now, you changed it by another equation – if I understand it right. Is $Q\neq0$ means here that the charge conservation does not hold? It's a very general formulation of the problem, is that really what you solving for? For example, take a look at this paper, where they model plasmonic structures and couple $\vec{J}$ and $\{\vec{E},\vec{H}\}$ in a certain way by using a variation of Yee's scheme. – Anton Menshov Jul 12 '18 at 22:38
• Thanks for the paper, I'll take a look at it! Yes, $Q \neq 0$ means that we have a source/sink. You're right, I am solving for a situation in which $\vec{v} \propto \vec{E} \times \vec{B}-\vec{E}-\vec{B}$, and $Q=const \neq 0$, $\vec{J} = \rho \vec{v}$ – justabear Jul 12 '18 at 22:57
• There are also complementary schemes (particle-in-cell or PIC) that treat charge(s) as free-moving discrete particles, as opposed to a continuous J field like Anton's reference. These methods are well suited for modeling particle beams, plasmas, and high power microwave sources (cyclotrons, magnetrons, etc). There's a good survey article here that you can mine for more info: Meierbachtol, Collin S., et al. "Conformal electromagnetic particle in cell: A review." IEEE Transactions on Plasma Science 43.11 (2015): 3778-3793. – rchilton1980 Jul 13 '18 at 0:18
• There are a bunch of papers similar to the one mentioned by Anton. You can search the key word (quantum) hydrodynamic model and plasmonics. – Pu Zhang Jul 13 '18 at 1:24