On the design of numerical methods (computational electromagnetics)

摘要

Many terms and ideas used in numerical methods have their origin in analytical mathematics. Despite the well-known discrepancies between number spaces of computers and those of mathematics, the consequences of applying mathematical theorems to numerical methods and the importance of physical reasoning are often underestimated. It is demonstrated that terms known from analytic considerations and goals like orthogonal basis functions and small condition numbers of matrices can be misleading, and can prevent engineers from designing useful codes for computational electromagnetics and similar tasks. Introducing a priori knowledge in numerical codes requires open structures, and often leads to ill-conditioned matrices. Thus, it is important to develop and apply methods for handling matrices such as the generalized point matching used in the multiple multipole (MMP) code instead of the projection technique used in many method of moments (MoM) codes.