Parabolic Equation Methods in Electromagnetics

Abstract

The parabolic equation (PE) has been solved by the finite difference schemes in the past decades for propagating and scattering problems. The Crank–Nicolson (CN) scheme is widely used for its simplicity and stability. By using theCNscheme, the parabolic equation can be solved in a sequential manner along the paraxial direction. In this way, a three‐dimensional problem can be converted into a series of two‐dimensional problems. Then the alternating direction implicit (ADI) scheme is introduced to solve the parabolic equation. It can be found that theADI‐PEmethod is more computationally efficient than theCN‐PEmethod, since the two‐dimensional problem can be further reduced to a series of one‐dimensional problems to be solved by theADI‐PEmethod. At last, the alternating group explicit (AGE) scheme, together with theADIscheme, is used to solve the parabolic equation. As a result, the unknowns in each transverse plane can be directly obtained without solving any matrix equations. Therefore, limited computing resources are needed even for the electrically large problems with good accuracy. Numerical results of both the propagating and scattering problems are given to verify the accuracy and efficiency of all the methods. The corresponding time‐domain counterparts are also introduced in this article.