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The phase error in finite-difference (FD) methods is related to the spatial resolution and thus limits the maximum grid size for a desired accuracy. Greater accuracy is typically achieved by defining finer resolutions or implementing higher order methods. Both these techniques require more memory and longer computation times. In this paper, new modified methods are presented which are optimized to problems of electromagnetics. Simple methods are presented that reduce numerical phase error without additional processing time or memory requirements. Furthermore, these methods are applied to both the Helmholtz equation in the frequency domain and the finite-difference timedomain (FDTD) method. Both analytical and numerical results are presented to demonstrate the accuracy of these new methods.

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A new finite difference method for the Helmholtz equation is presented. The method involves replacing the standard "weights" in the central difference quotients (Sects. 2.1, 2.2, and 2.3) by weights that are optimal in a sense that will be explained in the Sects. just mentioned. The calculation of the optimal weights involves some complicated and error prone manipulations of integral formulas that is best done using computer aided symbolic computation (SC). In addition, we discuss the important problem of interpolation involving meshes that have been refined in certain subregions. Analytic formulae are derived using SC for these interpolation schemes. Our results are discussed in Sect. 5. Some hints about the computer methods we used to accomplish these results are given in the Appendix. More information is available and access to that information is referenced. While we do not want to make SC the focus of this work, we also do not want to underestimate its value. Armed with robust and efficient SC libraries, a researcher can comfortably and conveniently experiment with ideas that he or she might not examine otherwise.

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