Simulation of Maxwell equation based on an ADI approach and integrated radial basis function-generalized moving least squares (IRBF-GMLS) method with reduced order algorithm based on proper orthogonal decomposition

Abstract

Basic equations of electromagnetic are Maxwell equations. In this manuscript, ADI-IRBF-GMLS is employed for solving the time-dependent Maxwell equations in two-dimension. For approximating the time variable, we utilize alternative direction implicit (ADI) method and integrated radial basis function based on generalized moving least squares (IRBF-GMLS) method is used for space direction. Alternative Direct Implicit (ADI) technique includes two steps in each time stage, that their computations are simple. We have to increase the number of collocation points and also time steps to reach the final time. This procedure increases the used execution time. To overcome this issue, we employ the proper orthogonal decomposition (POD) method to reduce the size of the final algebraic system of equations. This numerical procedure can be called ADI-IRBF-GMLS-POD method. Numerical results are presented and they illustrate the accuracy and efficiency of the proposed method. The point to note is that the used time-discrete scheme i.e. ADI approach cannot be employed in the numerical simulations on non-rectangular computational domains for solving Maxwell equations. This is the main issue in this numerical approach for Maxwell equations. We also compare the ADI-IRBF-GMLS with POD with full model of presented method applied to solve Maxwell equations.