Modelling Material Interfaces and Boundary Conditions in High-Order Finite-Difference Methods

Roberto Armenta

The Edward S. Rogers Sr. Department of Electrical and Computer Engineering (Electromagnetics)

Abstract:

When numerically solving electromagnetic problems using finite-difference methods, the enforcement of boundary conditions plays a dominant role in the performance of the overall algorithm. Finite-difference methods are created by substituting the partial derivatives in the differential form of Maxwell's equations with discrete approximations. Creating discrete finite-difference approximations of Maxwell's equations at material interfaces and at the outer boundaries of a given problem can be a challenging task since the employed approximations must enforce the appropriate boundary conditions correctly. The most important recent development in finite-difference methods has been the introduction of finite-difference approximations of variable accuracy. By using finite-difference approximations of variable accuracy, it is possible to increase the accuracy of the numerical results without making the discretisation grid finer. In most computer architectures the ability to change the accuracy of the employed finite-difference approximations can considerably improve the performance of the overall algorithm. The use of variable or high-order approximations was originally proposed in the late 1990s, and, initially, all the associated techniques to enforce boundary conditions were developed using exclusively uniform rectangular grids. Until fairly recently it was not clear how to enforce boundary conditions with a high-order of accuracy even on non-uniform one-dimensional grids. The reason behind this is that the role that the grid structure plays in the enforcement of boundary conditions was not clearly understood. In this seminar this role will be examined using simple one-dimensional transmission-line structures. In the process two newly developed methods for enforcing impedance boundary conditions and enforcing the continuity of the voltage and current at material interfaces will be introduced. To make the discussion accessible to everyone, everything will be explained starting from the transmission-line equations as introduced in most undergraduate textbooks. By the end of the seminar, the attendees should be able to understand why the term *high-order* is so often used in the advertisement of commercial packages.

Biography:

Roberto Armenta is a PhD student in the Electromagnetics group under the supervision of Prof. Costas D. Sarris and is working on the numerical solution of Maxwell's equations.