A newly modified method for 3D Maxwell's equation

ADI method is a traditional and effective numerical method, it has a lot of applications. ADI method has already been used to solve Maxwell's equations in vacuum, and its energy equation has been obtained. But when ADI method is applied to solve Maxwell's equations in other media, its energy equation temporarily can't be obtained. To obtain energy equation of unconditionally stable scheme, we first introduced energy equation of the conditionally stable leapfrog schemes for 3D Maxwell's equations in vacuum. We named the leapfrog schemes as the pre-modified schemes and its energy equation named as the pre-modified energy equation. Secondly, the modified energy equation was obtained by using the modification of pre-modified energy equation. Considering the modified energy equation can't match pre-modified schemes. Thus, the modified schemes was reversed by adding the modified term from pre-modified energy equation. Furthermore, we proposed the general modified scheme with α coefficient, and it can be adapted to more possibilities. We performed the stability and error estimate of the general modified methods. Numerical results have been achieved to verify second-order convergence rate in both time and space, and energy-preserving, regardless of the value of α.