A scheme-fitted splitting method for source terms to solve inhomogeneous Maxwell equations in time domain

In this article, we construct a family of solutions to nonhomogeneous Maxwell equations for which numerical computations exhibit an unphysical behavior for a number of tested schemes (FDTD, FVTD, and several DGTD schemes), the current density source term being discretized like the electrical field. We then propose a new, scheme-fitted method to account for this source term, resulting in the correction of the forementioned numerical results for all tested schemes. Doing so provides a new insight on test cases used in the framework of divergence cleaning techniques. The proposed method is based on Helmholtz Theorem. A general theoretical result, dictated from considerations at a continuous level, is given as a hint to explain the drastic and indiscriminate improvement numerically observed.