# Operator splitting for Maxwell's equations

In this book I've read about operator splitting method for solving differential equations. I for example I have an equation in the form $$\dot \omega (t) = A \omega(t)$$, where $A$ is some operator then I can split $A=A_1 + A_2$. The exact solution of the equation if I discretize in time will be $$\omega(t_{n+1}) = e^{\tau A}\omega(t_n)$$, and if I split $A$ then I can approximate the solution with $$\omega_{n+1} = e^{\tau A_2}e^{\tau A_1}\omega_n$$ with $\omega_n$ approximating $\omega(t_n)$. Now I've split the original equation in two equations:

$$\dot{\omega}^*(t) = A_1 \omega^*(t) \text{ for } t_n \le t \leq t_{n+1} \text{ with }\omega^*(t_n) = \omega_n$$ $$\dot{\omega}^{**}(t) = A_2 \omega^*(t) \text{ for } t_n \le t \leq t_{n+1} \text{ with }\omega^{**}(t_n) = \omega^*(t_{n+1})$$.

Ok after this math what I want to do is I want to rewrite Maxwell's equations in the form that is suitable for this method. I'll take the curls parts for the moment. And write them in the form

$$\partial _t \vec{F} + c \partial_x \vec{F} + c\partial_y B\vec{F} + c\partial_z \vec{F} = - \vec{J}$$ where $\vec{F} = (E_x, E_y, E_z, B_x, B_y, B_z)^T$ and $\vec{J} = (J_x, J_y, J_z, 0 , 0, 0)^T$. So this form is very similar to the one in operator splitting but now the equation is not homogenous and am not sure how would I solve this. Because I am solving this equations numerically I do not know how hard would it be to find particular solution and then add it to the homogenous one. I've search a lot for solving Maxwell's equations using this method and it is always they solve source free equations and that is not what I am interested in.

• For numerical implementation I highly recommend working with the vector potential directly; that way you get a four dimensional problem. The most compact while complete form of the inhomogeneous maxwell equations is $\partial_\mu F^{\mu\nu}=\mu_0 j^\nu$. Those equations take the form of coupled partial differential equations (sometimes possion-like) and are usally solved using spectral methods or green's functions. I have not seen a numerical aproach using that operator splitting you presented. – N0va Sep 6 '16 at 10:47
• Yeah that is the problem, because the only reference is Sentoku's PICLS code that use directional splitting how he calls it, but it is actually just operator splitting. I am trying to implement his directional splitting into another PIC code and there is not any reference about that. – Aleksandar Bukva Sep 6 '16 at 11:27