We have a 2D electromagnetic field (in the sense that: $E=(E_x,E_y,0)$, $B=(0,0,B_z)$, and all derivatives with respect to $z$ are $0$), and we are considering a system made up of two walls at $x=-b$ and $x=b$ held at fixed potential $\phi(y)=e^{-y}$, with a third conducting wall at $y=0$ held at $\phi(x)=1$. The current in our system is $\vec{j} \propto \nabla \cdot E\left(B \times E+\sigma E \right)$. The initial conditions are that $E=-\nabla \phi$, where $\phi$ is the solution to the Laplace equation $\nabla^2 \phi = 0$, and $B=0$.

Yee's scheme is resulting in diverging $E_y$, so what numerical scheme would you suggest be used to solve Maxwell's equations in this case?


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