Reviews | Contents |
Durran, Dale R.
    Numerical methods for wave equations in  geophysical fluid dynamics. - New York etc.: Springer, cop.1999. - xvii, 465 p. : ill. ; 25 cm. (Texts in applied mathematics; 32). - Includes bibliographical references (p. [443]-455) and index.
ISBN: 0-387-98376-7 (New York : alk. paper)   KV99/3884    : 784.00
fluid dynamics -- methodology
geophysics --methodology
wave equation
numerical analysis
differential equations, partial -- numerical solutions

REVIEW

From The Publisher:
     This scholarly text provides an introduction to the numerical methods used to model partial differential equations governing wave-like and weakly dissipative flows. The focus of the book is on fundamental methods and standard fluid dynamical problems such as tracer transport, the shallow-water equations, and the Euler equations. The emphasis is on methods appropriate for applications in atmospheric and oceanic science, but these same methods are also well suited for the simulation of wave-like flows in many other scientific and engineering disciplines. Numerical Methods for Wave Equations in Geophysical Fluid Dynamics will be useful as a senior undergraduate and graduate text, and as a reference for those teaching or using numerical methods, particularly for those concentrating on fluid dynamics.

 From Acta Meteorologica Sinica - Ding Yihu, Zhao Nan, and Zhou Jiangxing:
     ...an excellent reference in computational fluid dynamics, and numerical methods in applied partial differential equations. This book should be brought to the attention of readers in meteorology, oceanography, physics, mechanics and engineering. It is an excellent textbook. This book gives extensive and deep discussions of the main theoretical and practical research results in numerical geophysical fluid dynamics. Compared with other books in this field, the extensiveness, theoretical and practical deepness this book reaches makes it a classical work.

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CONTENTS

1 Introduction
    1.1 Partial Differential Equations---Some Basics
           First-Order Hyperbolic Equations
           Linear Second-Order Equations in Two Independent Variables
    1.2 Wave Equations in Geophysical Fluid Dynamics
           Hyperbolic Equations
           Filtered Equations
     1.3 Strategies for Numerical Approximation
           Approximating Calculus with Algebra
           Marching Schemes
           Problems
2 Basic Finite-Difference Methods
     2.1 Accuracy and Consistency
     2.2 Stability and Convergence
           The Energy Method
           Von NeumannÆs Method
           The Courant-Fredrichs-Lewy Condition
     2.3 Time-Differencing
           The Oscillation Equation, Phase Speed and Amplitude Error
           Single-Stage Two-Level Schemes
           Multi-Stage Methods
           Three-Level Schemes
           Controlling the Leapfrog Computational Mode
           Higher Order Schemes
     2.4 Space Differencing
           Differential-Difference Equations and Wave Dispersion
           Dissipation, Dispersion and the Modified Equation
           Artificial Dissipation
           Compact Differencing
     2.5 Combined Time and Space Differencing
           The Discrete Dispersion Relation
           The Modified Equation
           The Lax-Wendroff Method
     2.6 Summary Discussion of Elementary Methods
           Problems
 3 Beyond the One-Wave Equation
     3.1 Systems of Equations
           Stability
           Staggered meshes
     3.2 Three or more independent variables
           Scalar Advection in Two Dimensions
           Systems of equations in several dimensions
     3.3 Splitting into Fractional Steps
           Split explicit schemes
           Split implicit schemes
           Stability of split schemes
     3.4 Diffusion, Sources and Sinks
           Pure Diffusion
           Advection and Diffusion
           Advection with Sources and Sinks
     3.5 Linear Equations with Variable Coefficients
           Aliasing error
           Conservation
     3.6 Nonlinear Instability
           BurgersÆ equation
           The barotropic vorticity equation
           Problems
 4 Series-Expansion Methods
     4.1 Strategies for Minimizing the Residual
     4.2 The Spectral Method
           Comparison with Finite-Difference Methods
           Improving Efficiency Using the Transform Method
           Conservation and the Galerkin Approximation
     4.3 The Pseudospectral Method
     4.4 Spherical Harmonics
           Truncating the Expansion
           Elimination of the Pole Problem
           Gaussian Quadrature and the Transform Model
           Nonlinear Shallow-Water Equations
     4.5 The Finite Element Method
           Galerkin Approximation with Chapeau Basis Functions
           Quadratic Basis Functions
           Cubic Basis Functions
           Finite Elements on Rectangles
           Non-Rectangular Domains
           Problems
 5 Finite Volume Methods
     5.1 Conservation Laws and Weak Solutions
           The Riemann problem
           Entropy-consistent solutions
     5.2 Finite-Volume Methods and Convergence
           Monotone Schemes
           TVD Methods
     5.3 Discontinuities in Geophysical Fluid Dynamics
     5.4 Flux-Corrected Transport
           Flux Correction: The Original Proposal
           The Zalesak Corrector
           Iterative Flux Correction
     5.5 Flux Limiter Methods
           Insuring that the Scheme is TVD
           Possible Flux Limiters
           Flow Velocities of Arbitrary Sign
     5.6 Approximation with Local Polynomials
           GodunovÆs Method
           Piecewise-Linear Functions
     5.7 Two Spatial Dimensions
           FCT in Two-Dimensions
           Flux-Limiter Methods for Uniform 2D Flow
           Non-Uniform Non-Divergent Flow
           A Numerical Example
           When is a Flux-Limiter Necessary?
     5.8 Schemes for Positive-Definite Advection
           An FCT Approach
           Anti-Diffusion via Upstream Differencing
     5.9 Curvilinear Coordinates
           Problems
6 Semi-Lagrangian Methods
     6.1 The Scalar Advection Equation
           Constant Velocity
           Variable Velocity
     6.2 Forcing in the Lagrangian Frame
     6.3 Systems of Equations
           Comparison with the Method of Characteristics
           Semi-Implicit Semi-Lagrangian Schemes
     6.4 Alternative Trajectories
           A Non-Interpolating Leapfrog Scheme
           Interpolation via Parameterized Advection
     6.5 Eulerian or Semi-Lagrangian?
           Problems
7 Physically Insignificant Fast Waves
     7.1 The projection method
           Forward-in-Time Implementation
           Leapfrog Implementation
           Solving the Poisson Equation for Pressure
     7.2 The Semi-Implicit Method
           Large time steps and poor accuracy
           A prototype problem
           Semi-implicit solution of the shallow-water equations
           Semi-implicit solution of the Euler equations
           Numerical Implementation
     7.3 Fractional step methods
           Complete operator splitting
           Partially-split operators
     7.4 Summary of Schemes for Nonhydrostatic Models
     7.5 The Hydrostatic Approximation
     7.6 Primitive Equation Models
           Pressure and &sgr; coordinates
           Spectral Representation of the Horizontal Structure
           Vertical Differencing
           Energy Conservation
           Semi-Implicit Time Differencing
           Problems
 8 Non-reflecting Boundary Conditions
     8.1 One-dimensional flow
           Well-posed initial-boundary value problems
           The radiation condition
           Time-Dependent Boundary Data
           Reflections at an artificial boundary---the continuous case
           Reflections at an artificial boundary---the discretized case
           Stability in the presence of boundaries
     8.2 Two-dimensional shallow-water flow
           One-way wave equations
            Numerical implementation
     8.3 Two-dimensional stratified flow
           Lateral boundary conditions
           Upper boundary conditions
           Numerical implementation of the radiation upper boundary condition
     8.4 Wave-absorbing layers
     8.5 Summary
           Problems
           A Numerical Miscellany
     A.1 Finite-Difference Operator Notation
     A.2 Tridiagonal Solvers
     Code for a Tridiagonal Solver
     Code for a Periodic Tridiagonal Solver
     Bibliography

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MT (marje@cs.ioc.ee) 18.08.99