Error characterization for the time-domain numerical solution of Maxwell´s equations

The accuracy and stability of transient numerical methods in electromagnetics is an important consideration, for which an analytical measure is necessary. Of the several approaches commonly used to indicate accuracy of numerical methods, the modified equation of Warming and Hyett [1974] is examined, for several finite-difference and finite-element time-domain methods. In the method of Warming and Hyett, a Taylor-series representation of each unknown is substituted into the difference equation and, through a rather lengthy series of algebraic steps, reduced to what is known as a modified equation. From this modified equation, which represents the actual differential equation being solved by the numerical method, the stability, and relative phase and magnitude accuracy, of the method can be ascertained. As a final introductory note, the one-dimensional equations for each of the algorithms are used in the development of the modified equation and the dispersion relations. The extension of each of the one-dimensional results to two and three dimensions is discussed, in the context of each method.<>