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Lecture Notes - Masthead
 
Numerical Methods in the Hydrological Sciences
by George Hornberger and Patricia Wiberg
 

Table of Contents

  Title Page  
  Copyright Page  
i
Preface  
iii
Introduction  
v
About the Authors  
1.
Computation with MATLAB  
1.1.
Data in MATLAB
The basics: Matrices and vectors
Setting vectors and matrices with internal MATLAB commands
Simple functions and plots
Reading data into MATLAB
 
1.2.
Mathematical operations with matrices
Elementary operations: Addition, subtraction, and so forth
Matrix multiplication
Multiplication on an element-by-element basis
 
1.3.
1.4.
Symbolic math toolbox
Programming in MATLAB
Script files
Function files
 
1.5.
1.6.
1.7.
Summary
Problems
References
 
2.
 
2.1
2.2.
2.3.
Nonlinear algebraic equations
Bisection
Newton's method and secant method
Newton's method
Secant method
 
2.4.
2.5.
2.6.
2.7.
2.8.
2.9.
2.10.
2.11.
2.12.
2.13.
2.14.
MATLAB methods for finding roots
Example: Leonardo of Pisa’s polynomial
Iterative equations
MATLAB methods for solving iterative equations
Example: Wavelength of surface gravity waves
Systems of linear equations
MATLAB methods for solving sets of linear equations
Gaussian elimination
Example: Gaussian elimination
Problems
References
 
3.
 
3.1.
3.2.
3.3.
3.4.
3.5.
3.6.
3.7.
3.8.
3.9.
3.10.
3.11.
Finite differences
Taylor series approach
Differentiation using interpolating polynomials
MATLAB methods for finding derivatives
Numerical integration
Trapezoidal rule
Simpson's rules
Gaussian quadrature
MATLAB methods
Problems
References
 
4.
 
4.1.
4.2.
4.3.
4.4.
4.5.
4.6.

Ordinary differential equations in hydrology
Euler and Taylor-series methods
Modified Euler method or Euler predictor-corrector method
Runge-Kutta methods
Runge-Kutta-Fehlberg method
MATLAB methods
Example
 
4.7.
4.8.
4.9.
4.10.
4.11.
Multistep methods
Sources of error, convergence, and stability
Systems of equations
Problems
References
 
5.
 
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
5.7.
Introduction
Solving higher-order initial value problems
Example: Bessel equation
Solving boundary value problems using the shooting method
Solving boundary value problems using finite differences
Derivative boundary conditions
Problems
 
6.
 
6.1.
Classification of partial differential equations  
6.2.
Finite difference approximations for derivatives  
6.3.
Example: The Laplace equation
Solving the example problem in MATLAB
 
6.4.
6.5.
6.6.
6.7.
6.8.
Example problem: The "Tóth" problem
Solving the two-dimensional Poisson equation
An example problem
Problems
References
 
7.
 
7.1.
7.2.
7.3.
7.4.
7.5.
7.6.
7.7.
Introduction
Fixed-point iteration revisited
Iterative solution of a system of equations
Vector and matrix norms
Iterative methods for finite difference equations
Problems
References
 
8.
 
8.1.
8.2.
8.3.
8.4.
8.5.
8.6.
8.7.
8.8.
8.9.
Background
Unidirectional flow in an aquifer
A forward-difference (or explicit) approximation
Stability problems with the explicit method
A backward-difference (or implicit) approximation
The g method
Flow in an unsaturated soil
Problems
References
 
9.
Finite Difference Methods for Transport Equations  
9.1.
9.2.
9.3.
9.4.
9.5.
9.6.
Background
Numerical dispersion
The advection-dispersion equation
Transport of reactive solutes
Problems
References
 
10.
The Finite Element Method: An Introduction  
10.1.
10.2.
10.3.
10.4.
10.5.
10.6.
10.7.
Background
Collocation
Weighted residual method
The finite-element approach: Galerkin weighted residual method
Steady diffusion into sediment
Problems
References
 
11.
The Finite Element Method: Steady Flow of Groundwater in Two Dimensions  
11.1.
11.2.
11.3.
11.4.
11.5.
11.6.
11.7.
11.8.
11.9.
11.10.
Background
An example problem
Triangular elements
The weighted residual equation and its integral over an element
Defining node connectivities
Assigning x and z coordinates to the mesh
Assembling the global conductance matrix
The boundary conditions: Setting the right-hand side of the equation
Problems
References
 
12.
Methods of Data Analysis: Fourier Analysis of Time Series  
12.1.
12.2.
12.3.
12.4.
12.5.
12.6.
12.7.
12.8.
12.9.
Periodic functions
Fourier series
Example: Square wave
Example: CO2 time series
MATLAB methods
Spectral analysis and periodograms
Filtering time series
Problems
References
 
13.
Methods of Data Analysis: Spatial Data  
13.1.
13.2.
13.3.
13.4.
13.5.
13.6.
13.7.
Background
Interpolating irregularly spaced data onto a regular grid
Contouring
Semivariance
Kriging
Problems
References
 
Subject Index  


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