Symplectic Electromagnetic Scattering Numerical Methods

Computational electromagnetics have been developed rapidly with the development of the compute hardware and software, and various numerical methods for electromagnetic field simulation are proposed in years. But these methods have difficulty in the aspects of computational time, required memory and accuracy. In addition, with the depth of the acquaintance with the physical problems, people realize that it is also important to keep some features of the system while they pursue the accuracy of the algorithm. Since the linear or nonlinear electromagnetic field equations can be written as an infinite-dimensional Hamiltonian system, whose solution can be viewed as a Hamiltonian flow in the phase space which preserves the symplectic structure in the time direction. Such important features should not be neglected during the construction of numerical methods for the field equations. The symplectic integrator method is the new time-domain method which is specialized to a Hamiltonian system and can preserve the symplectic structure of the phase space. It shows substantial benefits in computational accuracy, time, etc. The dissertation mainly focuses on the basic derivation and application of this method. The main studies are as follows:(1) The Hamiltonian mechanics and equations are deduced from the Lagrange mechanics. The symplectic quality of the Hamiltonian system is discussed. The formulations of the symplectic integrator method are constructed, especially the explicit symplectic schemes for the separable Hamiltonian system and the symplectic partitioned Runge-Kutta (PRK) method for the generic Hamiltonian system.(2) The symplectic integrator method is applied to the high frequency asymptotic evolution of the wave equation based on Maslov symplectic geometrical theory. Maslov's method can overcome the drawback that the solution in the caustic region cannot be obtained with geometrical optics.The key step of the method is numerical computation of Hamiltonian equations, which can be solved by the symplectic algorithm. The principle of the method is illuminated by the numerical simulation of the scattering of caustic region in a two-dimensional concave antenna.(3) The symplectic integrator method is applied to time evolution of the scalar wave equation. Since the wave equation can be formulated as a Hamiltonian system, the explicit symplectic method is applicable to the time-domain simulation for the wave equation. The accuracies of the different orders symplectic difference schemes are compared and the effect of the spatial discretization methods upon the accuracy is analyzed by simulating the propagation of a one-dimensional wave under the periodic boundary condition.(4) The last, the symplectic algorithm is applied to the time-domain simulation for Maxwell Equations. The author focuses discussion on Maxwell Equations in two dimensions involving respectively the two conditions that the current density is zero and the current density exists. The symplectic integrator method is implicit except that the Hamiltonian is separable. But when the current density exists, the Hamiltonian is not separable. Here the author demonstrates the symplectic PRK method is still applicable, and the schemes are still explicit. The accuracy, computational time and required memory of the symplectic PRK method are compared with the standard FDTD method. The results show that the symplectic integrator method is promising.