Unconditionally stable meshless integration of time-domain Maxwell's curl equations

Grid based methods coupled with an explicit approach for the evolution in time are traditionally adopted in solving PDEs in computational electromagnetics. The discretization in space with a grid covering the problem domain and a stability step size restriction, must be accepted. Evidence is given that efforts need for overcoming these heavy constraints. The connectivity laws among the points scattered in the problem domain can be avoided by using meshless methods. Among these, the smoothed particle electromagnetics, gives an interesting answer to the problem, overcoming the limit of the grid generation. In the original formulation an explicit integration scheme is used providing, spatial and time discretization strictly interleaved and mutually conditioned. In this paper a formulation of the alternating direction implicit scheme is proposed into the meshless framework. The developed formulation preserves the leapfrog marching on in time of the explicit integration scheme. Studies on the systems matrices arising at each temporal step, are reported referring to the meshless discretization. The new method, not constrained by a grid in space and unconditionally stable in time, is validated by numerical simulations.