Efficient algorithm for proper-time computations of long electromagnetic phenomena in a bounded computational domain

Nonreflecting artificial boundary conditions (ABCs) and an efficient algorithm for their implementation are constructed, which replace difference Maxwell equations in a vacuum solved in proper time in a region surrounding the computational domain. One time step of the algorithm requires $O(T_f^1h^{-4})$ time rather than $O(T_f^4h^{-4})$ time taken in the direct computation without using ABCs. This makes it possible to compute certain electrodynamic problems at long times.