ABSTRACT
Maxwell’s equations are PDEs in time domain. Their solution with appropriate initial and boundary conditions give results for many practical problems in classical electrodynamics. Maxwell’s equations are supplemented by the “constitutive relations” to describe the properties of the medium. In previous chapters, we explored such solutions using numerical methods. Most of the examples involved static or time-harmonic solution. The corresponding PDEs are called elliptic type. We explored their solution using £nite differences. The technique consisted of replacing the continuous domain by a discrete set of points, which are the intersection points of a grid superposed on the continuous domain. The differential equation is converted into a set of algebraic equations by expressing the derivatives in terms of the potential at the neighboring points of the grid. This technique resulted in a set of simultaneous algebraic equations. The boundary of the domain is a closed one and the boundary conditions are the values of the potential or its functions on the boundary. The dif£culty is the slow convergence of the solution if iterative methods are used. If the simultaneous equations are solved by matrix inversion, then it could take a large amount of computer time. Moreover, due to šoating point arithmetic used by the computers, the round off error involved in performing a large number of arithmetic operations can overwhelm the actual solution leading to large errors. The waveguide problem we solved in the previous chapters is based on the Helmholtz equation which is an elliptic type PDE. However, the problem involved determination of the eigenvalues.
