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Preface
The desire to write this book has stemmed from the realization of the undeniable ben-
efits of formalizing, organizing, and cataloging material that I have developed and,
to some extent, collected, for more than a decade as part of an annual ritual – that of
teaching an entry-level graduate course entitled “Intermediate Numerical Methods.”
Through this book, I also intend to pay tribute to many great teachers who taught me
the fundamentals of numerical methods in graduate school, but have never had the
time or inclination to put their deep knowledge of the subject in writing.
Tremendous advancement in computer hardware over the past two decades has
resulted in the proliferation of scientific and engineering analysis, and powerful user-
friendly tools that enable such analysis. Modeling and simulation has now permeated
the industry and is slowly beginning to occupy a position of importance in the design
cycle of many products. Students graduating today with a degree in engineering or
applied sciences are almost mandated by their recruiters to have a working knowl-
edge of the preferred analysis tool in their discipline.
At the heart of many engineering and scientific analyses is the solution of differ-
ential equations – both ordinary and partial differential equations (PDEs). The solu-
tion of the latter type of equation can be very challenging, depending on the type of
equation, the number of independent variables, the boundary and initial conditions,
and other factors. A variety of broadly applicable methods have been developed to
this end. Among the deterministic methods for solving differential equations, the
most popular ones are the finite element method, the finite difference method, and the
finite volume method. Each method has its own pros and cons, and shines for a cer-
tain class of problems, for reasons that are deeply rooted in the mathematical founda-
tion of the method. Although trends are slowly changing, the finite element method
has been traditionally used for solving problems in solid mechanics, while the finite
difference and finite volume methods have been traditionally used to solve problems
involving fluid flow and heat transfer. These boundaries, though, are strictly defined
by history, not by the underlying mathematics.
Although this book is supposedly a general book on numerical methods for solv-
ing PDEs, and therefore, quite rightly, is expected to cover all relevant topics, certain
practical constraints, such as the size of the book, as well as assessment of what mate-
rial exists in other texts, has prompted me to exclude many topics. For example, I have
chosen to adhere to deterministic methods only, and stochastic methods for solving
PDEs have been excluded. Perhaps, the most important topic that has been excluded
is the finite element method. This important decision was prompted by the fact that
the finite element method has a deep-rooted history, and several well-written, high-
quality texts already exist on this topic. Thus, in this book, I have chosen to focus
on the finite difference and finite volume methods. In my quest for finding a book
that adequately covers all aspects of the finite difference and finite volume methods,
I have found that either the coverage is incomplete or too primitive. For example,
