| 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 |
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.
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