Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI

by White; Robert E.

• ISBN13: 9781584883647
• ISBN10: 1584883642
• Format: Hardcover
• Copyright: 2003-09-17
• Publisher: Chapman & Hall/

Summary

Computational Mathematics: Models, Methods, and Analysis with MATLAB and MPI explores and illustrates this process. Each section of the first six chapters is motivated by a specific application. The author applies a model, selects a numerical method, implements computer simulations, and assesses the ensuing results. These chapters include an abundance of MATLAB code. By studying the code instead of using it as a "black box, " you take the first step toward more sophisticated numerical modeling. The last four chapters focus on multiprocessing algorithms implemented using message passing interface (MPI). These chapters include Fortran 9x codes that illustrate the basic MPI subroutines and revisit the applications of the previous chapters from a parallel implementation perspective. All of the codes are available for download from www4.ncsu.edu./~white.This book is not just about math, not just about computing, and not just about applications, but about all three--in other words, computational science. Whether used as an undergraduate textbook, for self-study, or for reference, it builds the foundation you need to make numerical modeling and simulation integral parts of your investigational toolbox.

Author Biography

Robert E. White is a Professor in the Department of Mathematics at North Carolina State University, Raleigh, USA.

Table of Contents

List of Figures	p. ix
List of Tables	p. xi
Preface	p. xiii
Introduction	p. xv
Discrete Time-Space Models	p. 1
Newton Cooling Models	p. 1
Heat Diffusion in a Wire	p. 9
Diffusion in a Wire with Little Insulation	p. 17
Flow and Decay of a Pollutant in a Stream	p. 25
Heat and Mass Transfer in Two Directions	p. 32
Convergence Analysis	p. 42
Steady State Discrete Models	p. 51
Steady State and Triangular Solves	p. 51
Heat Diffusion and Gauss Elimination	p. 59
Cooling Fin and Tridiagonal Matrices	p. 68
Schur Complement	p. 77
Convergence to Steady State	p. 86
Convergence to Continuous Model	p. 91
Poisson Equation Models	p. 99
Steady State and Iterative Methods	p. 99
Heat Transfer in 2D Fin and SOR	p. 107
Fluid Flow in a 2D Porous Medium	p. 116
Ideal Fluid Flow	p. 122
Deformed Membrane and Steepest Descent	p. 130
Conjugate Gradient Method	p. 138
Nonlinear and 3D Models	p. 145
Nonlinear Problems in One Variable	p. 145
Nonlinear Heat Transfer in a Wire	p. 152
Nonlinear Heat Transfer in 2D	p. 159
Steady State 3D Heat Diffusion	p. 166
Time Dependent 3D Diffusion	p. 171
High Performance Computations in 3D	p. 179
Epidemics, Images and Money	p. 189
Epidemics and Dispersion	p. 189
Epidemic Dispersion in 2D	p. 197
Image Restoration	p. 204
Restoration in 2D	p. 213
Option Contract Models	p. 219
Black-Scholes Model for Two Assets	p. 228
High Performance Computing	p. 237
Vector Computers and Matrix Products	p. 237
Vector Computations for Heat Diffusion	p. 244
Multiprocessors and Mass Transfer	p. 249
MPI and the IBM/SP	p. 258
MPI and Matrix Products	p. 263
MPI and 2D Models	p. 268
Message Passing Interface	p. 275
Basic MPI Subroutines	p. 275
Reduce and Broadcast	p. 282
Gather and Scatter	p. 288
Grouped Data Types	p. 294
Communicators	p. 301
Fox Algorithm for AB	p. 307
Classical Methods for Ax = d	p. 313
Gauss Elimination	p. 313
Symmetric Positive Definite Matrices	p. 318
Domain Decomposition and MPI	p. 324
SOR and P-regular Splittings	p. 328
SOR and MPI	p. 333
Parallel ADI Schemes	p. 339
Krylov Methods for Ax = d	p. 345
Conjugate Gradient Method	p. 345
Preconditioners	p. 350
PCG and MPI	p. 356
Least Squares	p. 360
GMRES	p. 365
GMRES(m) and MPI	p. 372
Bibliography	p. 379
Index	p. 381
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