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Chapter 2:
Accuracy, Stability, and Convergence of Numerical Algorithms
Many real phenomena are too complicated to comprehend, but we can attempt to understand them by constructing models that contain their essential features while ignoring minor complications. (What is essential is not always obvious, and the choice may depend on one’s purpose.) Models can yield deep insight into the nature of reality. A good model not only explains a phenomenon, but also predicts things yet to be observed. The quark model was devised in the early 1960s to simply explain the puzzling multiplicity of hadrons (that were being discovered) in terms of a few simple (presumed to be imaginary) elementary particles called quarks. No one believed that quarks actually existed, but later it was found that they do exist. It is remarkable indeed that the best model seems to be the simplest one that accounts for the facts - the principle of Occam’s razor. The FDTD methodology, the main subject of this book, is appealing because of its simplicity. Methodologies such as Feynman diagrams can take on lives of their own and even indicate new phenomena. A finite difference model of the logistic equation, which gives the wrong solution, has been used to study chaos (see Section 2.6.1). It has even been proposed that reality itself is just a set of algorithms. Indeed, in this book we will encounter algorithms meant to solve one problem that lead to solutions and models of other problems that explain - at least heuristically - certain real physical phenomena. Before embarking on the main topic of this book, we introduce a few example algorithms and analyze their accuracy and numerical stability. Our purpose is not only to show what can go wrong in computer calculations and how to avert it, but also to illustrate the physical insights that some of these algorithms yield. Topics • Numerical instability and chaos • Accuracy • Nonlinear problems • Parallel versus serial programs • Matrix analysis of a relaxation algorithm • Noninteger-order integration and differentiation
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