A meshless local Petrov–Galerkin method for the time-dependent Maxwell equations

In this paper, the meshless local Petrov–Galerkin (MLPG) method is employed to solve the 2-D time-dependent Maxwell equations. The MLPG method is a truly meshless method in which the trial and test functions are chosen from totally different functional spaces. In the current work, the moving least square reproducing kernel (MLSRK) scheme is chosen to be the trial function. The method is applied for the unsteady Maxwell equations in different media. In the local weak form, by employing the difference operator for evolution in time and simultaneously in time and space, the semi-discrete and fully discrete schemes are obtained respectively. The error estimation is discussed for both the semi-discrete and fully-discrete numerical schemes for modelling the time-dependent Maxwell equations. We show that provided that the time step size ττ is sufficiently small, the proposed scheme yields an error of O(ρ2(m+1)+τ2)O(ρ2(m+1)+τ2) in the L2L2 norm for the square of error. The new scheme is implemented and the numerical results are provided to justify our theoretical analysis.