Preface
This book is designed to serve as a textbook for graduate students or advanced
undergraduates studying numerical methods for the solution of partial differential equations goveming wave-like flows. Although the majority of the schemes
presented in this text were introduced in either the applied-rnathematics or atmos -
pheric-science literature, the focus is not on the nuts-and-bolts details of various
atmospheric models but on fundamental numerical methods that have applications
in a wide range of scientific and engineering disciplines. The prototype problems
considered include tracer transport, shallow-water flow and the evolution of internal waves in a continuously stratified fluid.
A significant fraction of the literature on numerical methods for these problems
falls into one of two categories, those books and papers that emphasize theorems
and proofs , and those that emphasize numerical experimentation. Given the uncertainty associated with the messy compromises actually required to construct
numerical approximations to real-world fluid-dynamics problems, it is difficult to
emphasize theorems and proofs without limiting the analysis to classical numerical schemes whose practical application may be rather limited . On the other hand,
if one relies primarilyon numerical experimentation it is much harder to arrive at
conclusions that extend beyond a specific set of test cases. In an attempt to establish a clear link between theory and practice, I have tried to follow amiddie
course between the theorern-and-proof formalism and the reliance on numerical
experimentation. There are no formal proofs in this book , but the mathematical
properties of each method are derived in a style familiar to physical scientists. At
the same time, numerical examples are includcd that illustrate these theoretically
derived properties and facilitate the intercomparison of various methods .