Durran D. Numerical methods for fluid dynamics: With applications to geophysics

Springer, 2010. — 533 p.
This book is a major revision of Numerical Methods for Wave Equations in Geophysical Fluid Dynamics; the new title of the second edition conveys its broader scope. The second edition is designed to serve graduate students and researchers studying geophysical fluids, while also providing a non-discipline-specific introduction to numerical methods for the solution of time-dependent differential equations. Changes from the first edition include a new Chapter 2 on the numerical solution of ordinary differential equations (ODEs), which covers classical ODE solvers as well as more recent advances in the design of Runge–Kutta methods and schemes for the solution of stiff equations. Chapter 2 also explores several characterizations of numerical stability to help the reader distinguish between those conditions sufficient to guarantee the convergence of numerical solutions to ODEs and the stronger stability conditions that must be satisfied to compute reasonable solutions to timedependent partial differential equations with finite time steps. Chapter 3 (formerly Chapter 2) has been reorganized and now covers finite-difference schemes for the simulation of one-dimensional tracer transport due to advection, diffusion, or both. Chapter 4, which is devoted to finite-difference approximations to more general partial differential equations, now includes an improved discussion of skew-symmetric operators. Chapter 5, Conservation Laws and Finite-Volume Methods, includes new sections on essentially nonoscillatory and weighted essentially nonoscillatory methods, the piecewise-parabolic method, and limiters that preserve smooth extrema. A section on the discontinuous Galerkin method now concludes Chapter 6, which continues to be rounded out with discussions of the spectral, pseudospectral, and finite-element methods. Chapter 7, Semi-Lagrangian Methods, now includes discussions of cascade interpolation and finite-volume integrations with large time steps. More minor modifications and updates have been incorporated throughout the remaining chapters.