A New Finite Difference Method for the Helmholtz Equation Using Symbolic Computation

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. 1.