More information about the book overall as well as supplementary material is available in its Web Companion: http://vmm.math.uci.edu/ODEandCM/ The web companion includes excerpts of each section. The first 38 pages of the chapter on design and analysis of computational methods are here.
In modern applied mathematics, theory and rigorously analyzed computation go hand-in-hand. This calls for a text that discusses both in detail, and we have undertaken to provide one. The discussion of numerical methods strives to be visual and comparative, based on carefully chosen examples of prototypical methods and model problems. For example, one-step and multistep methods are treated and compared side-by-side. This makes highlights the complementary challenges involved in understanding the behavior of both types of method when applied to particular problem; e.g., with multistep methods, the necessary conditions for a particular degree of accuracy are straightforward but the stability conditions required for convergence are subtle, while for one-step methods stability is automatic but the analysis of accuracy conditions is involved.
Figures illustrate the meaning of our example methods as well as the results of accuracy and stability studies performed with each method applied to a class of universal model problems, for example:
Some Example Methods (JPG, TIF)
Leapfrog Method Stability Study (JPG, TIF) Midpoint Method Stability Study (JPG, TIF) Trapezoidal Method Stability Study (JPG, TIF) Plane Rotation System Study (JPG, TIF)
The stability regions of our example methods are collected in one figure for comparison, and juxtaposed with visual representations of the spectra of important difference operators, e.g., forward, backward, centered difference and second difference, for example:
Absolute Stability Regions (JPG, TIF) Spectra of Difference Approximations (JPG, TIF)
When different ODE methods are used to approximate solutions of the two most important and universal initial value problems, the heat equation and the wave equation, the drastically different behavior that is observed is shown side-by-side, and explained as a consequence of the compatibility (or lack of compatibility) of the spectrum associated with the choice of differencing scheme and the absolute stability region of the various methods. Connections with the Fourier transform and the CFL condition are also treated in this context.
Diffusion equation, Euler's method (JPG, TIF) Advection Equation, Euler's method with Forward, Backward, and Centered Differencing (JPG, TIF) Advection Equation, Leapfrog method with Forward, Backward, and Centered Differencing (JPG, TIF)
The book also points out an inconsistency in the definition of the region of absolute stability of a numerical method in the literature. This fact is discussed in greater detail in the web companion, with examples from eight well-known books, four of which use one definition, and four of which use a conflicting definition. A direct link may be found here: http://vmm.math.uci.edu/ODEandCM/StabiltyRegionDefinitions/StabilityRegionDefinitions2.html
Rigorous convergence proofs are motivated by developing a strong intuition for the concepts involved. The meaning of different kinds of errors (e.g., local truncation error, residual errors, local error, global error) and also their relationships are depicted. The use of local error analysis for automatic step-size adjustment is handled with concrete examples that can be computed and understood exactly.
Local Solutions and Errors (JPG, TIF)
The conditions for a Runge-Kutta method to have a given degree of accuracy are analyzed visually using rooted trees as a tool. We show that the two expansions that must be matched to a given order correspond to building these trees in complementary ways. The terms on the Taylor expansion side are obtained from previous terms by the Leibniz rule combined with the chain rule. This corresponds to constructing new rooted trees by adding one edge and a leaf to each node of previously constructed trees. The terms on the Runge-Kutta expansion side are obtained from previous terms by performing a Taylor series expansion of a function whose argument is perturbed by a Taylor series. This corresponds to constructing new rooted trees by joining any number of previously constructed trees at a new root. This perspective, and the LISP notation we use to simplify the calculations, aims to introduce readers in a new and accessible way to the beautiful work of Butcher and others on Runge-Kutta methods.
The two streams of the book, the Theoretical and the Numerical, come together when results of the computational part of the book are applied to the Korteweg-deVries (KdV) equation. The numerical method we introduce to solve the KdV equation uses both the Fast Fourier Transform (FFT) and split-stepping.
The following is from the AMS description of the book at: http://www.ams.org/bookstore?fn=20&arg1=diffequ&ikey=STML-51
This book provides a conceptual introduction to the theory of ordinary differential equations, concentrating on the initial value problem for equations of evolution and with applications to the calculus of variations and classical mechanics, along with a discussion of chaos theory and ecological models. It has a unified and visual introduction to the theory of numerical methods and a novel approach to the analysis of errors and stability of various numerical solution algorithms based on carefully chosen model problems. While the book would be suitable as a textbook for an undergraduate or elementary graduate course in ordinary differential equations, the authors have designed the text also to be useful for motivated students wishing to learn the material on their own or desiring to supplement an ODE textbook being used in a course they are taking with a text offering a more conceptual approach to the subject.
