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.

Theoretical proofs state that the planar Winslow or homogenous Thompson-Thames-Mastin (hTTM) map is a diffeomorphism, yet numerical solutions to the hTTM equations produce folded grids on the so-called "horseshoe" domain. A quasi-analytic solution to the horseshoe problem is constructed to demonstrate that folding is due to truncation error effects. Higher-order difference methods are also explored. 0 1995 John Wiley & Sons. Inc.

The numerical solution of boundary value problems with curved domains in physics and engineering by conventional finite difference and finite element programs typically requires an enormous investment of time and effort by the user during the problem setup stage; an investment of several manweeks may be required to extract from engineering drawings a sufficiently high fidelity approximation to the problem domain in terms of a catenation of triangular, rectangular, and/or isoparametric elements or cells. Such tedious manual selection of an appropriate mesh for finite element or finite difference programs may, in fact, be necessary for highly complex structures; however, the authors describe their expermnces with the implementation of an alternative suggested by Gordon and Hall [4]. The discussion is restricted mainly to finite difference schemes, although finite element models can easily be developed along the same lines.

Advances in computer capabilities make the process of generating numerical solutions to complex partial differential equations fairly routine. At the same time, computer power now allows the direct manipulation of mathematical symbols and exact solutions. Symbolic computation engines now widely available include several successful commercial varieties like mathematica, maple, and matlab. Given this, it seems that the time is at hand to revisit the original problem of solving partial differential equations using a combination of numerical calculation and symbolic manipulation. Harnessing the power and potential of symbolic computation requires a reorientation of the computer and numerical analysts' perspective to achieve advances beyond current techniques. The new methodology emerging reopens the research field in computational fluid dynamics to possibilities only dreamed of in its infancy. Computer power at the time, however, forced the research direction towards fully discretized, purely numerical calculations. We present several examples of this emerging methodology in re-visiting some familiar, textbook problems. This will introduce the reader to the hybrid methodology with the hope of inspiring new thought in utilizing symbolic manipulation to solve problems in mathematical physics from a fundamental perspective.

With all of the experience and knowledge gained from symbolic and numeric computing in the last 20 years along with advances in software engineering and the advances in terms of the availability of powerful computers, it seems that the time is at hand to revisit the original problem of solving partial differential equations using a combination of numerical calculation and symbolic manipulation. We believe that hardware technology today has become powerful enough to realize the original dream of implementing mathematics at a high level on machines in robust and efficient ways. We used computer aided symbolic computation in our previous paper 1 to explore the problem of solving partial differential equations in a more general way. Here we revisit some classic techniques in light of new technology and extend the new methodology emerging here. Continued development of the hybrid methodology is presented with the hope of inspiring new thought in the creation of hybrid symbolic-numeric algorithms for solving the equations of mathematical physics from a fundamental perspective.