Initial value theorem

Abstract: The solution of Maxwell's equations in the time domain has now been in use for almost three decades and has had great success in many different applications. The main attraction of the time domain approach, originating in a paper by Yee (1966), is its simplicity. Compared with conventional frequency domain methods it takes only marginal effort to write a computer code for solving a simple scattering problem. However, when applying the time domain approach in a general way to arbitrarily complex problems, many seemingly simple additional problems add up. We describe a theoretical framework for solving Maxwell's equations in integral form, resulting in a set of matrix equations, each of which is the discrete analogue to one of the original Maxwell equations. This approach is called Finite Integration Theory and was first developed for frequency domain problems starting about two decades ago. The key point in this formulation is that it can be applied to static, harmonic and time dependent fields, mainly because it is nothing but a computer-compatible reformulation of Maxwell's equations in integral form. When specialised to time domain fields, the method actualy contains Yee's algorithm as a subset. Further additions include lossy materials and fields of moving charges, even including fully relativistic analysis. For amny practical problems the pure time domain algorithm is not sufficient. For instance a waveguide transition analysis requires knowledge of the incoming and outgoing mode patterns for proper excitation in the time domain. This is a typical example where both frequency and time domian analysis are essential and only the combinatin yields the successful result. Typical engineers may wonder why at all one should apply time domain analysis to basically monochromatic field problems. The answer is simple: it is much faster, needs less computer memory, is more general nad typically more accurate. Speed-up factors of over 200 have been reached for realistic problems in filter and waveguide design. The small core space requirement makes time domain methods applicable on desktop computers using milions of cells, and six unknowns per cell—a dimension that has not yet been reached by frequency domain approaches. This enormous amount of mesh cells is absolutely neceesary when complex structures or structures with spacial dimensions of many wavelengths are to be studied. Our personal recod so far is a waveguide problem in which we used 72,000,000 unknowns.