Springer Computational Mathematics

By: Willem Hundsdorfer, Jan G. Verwer

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Hardcover | 3 April 2007

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484 Pages

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23.5 x 15.88 x 3.18

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This book describes numerical methods for partial differential equations (PDEs) coupling advection, diffusion and reaction terms, encompassing methods for hyperbolic, parabolic and stiff and nonstiff ordinary differential equations (ODEs). The emphasis lies on time-dependent transport-chemistry problems, describing e.g. the evolution of concentrations in environmental and biological applications. Along with the common topics of stability and convergence, much attention is paid on how to prevent spurious, negative concentrations and oscillations, both in space and time. Many of the theoretical aspects are illustrated by numerical experiments on models from biology, chemistry and physics. A unified approach is followed by emphasizing the method of lines or semi-discretization. In this regard this book differs substantially from more specialized textbooks which deal exclusively with either PDEs or ODEs. This book treats integration methods suitable for both classes of problems and thus is of interest to PDE researchers unfamiliar with advanced numerical ODE methods, as well as to ODE researchers unaware of the vast amount of interesting results on numerical PDEs.

Industry Reviews

From the reviews:

"The numerical solution of time-dependent advection-diffusion-reaction problems draws on different areas of numerical analysis ... . We appreciate that the quite thorough, yet not pedantic, analytic part of the presentation is intimately interwoven with numerical tests and examples which will enable the reader to judge on the relative merits of the various approaches and really aid him in developing proper software for the problem at hand." (H. Mutsham, Monatshefte fuer Mathematik, Vol. 144 (2), 2005)

"Let me say at the outset that I highly recommend this book to practitioners ... end-users, and those new to the field. One of its strengths is its in-depth presentation of temporal and spatial discretizations and their interaction ... . With each topic, key theoretical results are presented. ... I found the present authors' choice of problems to be one of the highlights of the book." (Peter Moore, SIAM Review, Vol. 46 (3), 2004)

"This excellent research monograph contains a comprehensive discussion of numerical techniques for advection-reaction-diffusion partial differential equations (PDEs). The emphasis is on a method of lines approach, the analysis is careful and complete, and the numerical tests designed to verify the theoretical discussions of stability, convergence, monotonicity, etc. involve solving 'real life' equations. ... As is to be expected in such a carefully prepared monograph, there is an extensive bibliography and a good index. Highly recommended." (Ian Gladwell, Mathematical Reviews, 2004 g)

"The information, densely packed on roughly 450 pages, is abundant though well-structured, smoothly readable, and with emphasis on explanation of key concepts by means of examples that are stripped from unnecessary complications. ... a serious student with a hands-on attitude finds in this book an excellent source for self-studies and investigation. ... It is a valuable contribution to theSpringer Series in this field of research." (J. Brandts, Nieuw Archief voor Wiskunde, Vol. 7 (1), 2006)

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You Can Find This Book In

Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Engineering & Technology Technology in General Maths for Engineers Science Science in General Maths for Scientists Engineering in General

Numerical Analysis

Basic Concepts and Discretizations	p. 1
Advection-Diffusion-Reaction Equations	p. 1
Nonlinear Reaction Problems from Chemistry	p. 3
Model Equations for Advection-Diffusion	p. 9
Multi-dimensional Problems	p. 14
Examples of Applications	p. 18
Basic Discretizations for ODEs	p. 23
Initial Value Problems and Euler's Method	p. 23
Norms and Matrices	p. 27
Perturbations on ODE Systems	p. 30
The -Method and Stiff Problems	p. 35
Stability of the -Method	p. 37
Consistency and Convergence of the -Method	p. 42
Nonlinear Results for the -Method	p. 44
Concluding Remarks	p. 46
Basic Spatial Discretizations	p. 48
Discrete Fourier Decompositions	p. 49
The Advection Equation	p. 52
The Diffusion Equation	p. 62
The Advection-Diffusion Equation	p. 66
Convergence of Spatial Discretizations	p. 71
Stability, Consistency and Convergence	p. 71
Advection-Diffusion with Constant Coefficients	p. 74
Advection with Variable Coefficients	p. 77
Diffusion with Variable Coefficients	p. 81
Variable Coefficients and Higher-Order Schemes	p. 83
Boundary Conditions and Spatial Accuracy	p. 84
Refined Global Error Estimates	p. 85
Outflow with Central Advection Discretization	p. 86
Boundary Conditions with the Heat Equation	p. 88
Boundary Conditions and Higher-Order Schemes	p. 92
Time Stepping for PDEs	p. 94
The Method of Lines and Direct Discretizations	p. 94
Stability, Consistency and Convergence	p. 99
Stability for MOL - Stability Regions	p. 103
Von Neumann Stability Analysis	p. 111
Monotonicity Properties	p. 116
Positivity and Maximum Principle	p. 116
Positive Semi-discrete Systems	p. 118
Positive Time Stepping Methods	p. 121
Numerical Illustrations	p. 124
Numerical Test Examples	p. 127
The Nonlinear Schrödinger Equation	p. 128
The Angiogenesis Model	p. 134
Time Integration Methods	p. 139
Runge-Kutta Methods	p. 139
The Order Conditions	p. 140
Examples	p. 142
The Stability Function	p. 144
Step Size Restrictions for Advection-Diffusion	p. 149
Rosenbrock Methods	p. 151
Convergence of Runge-Kutta Methods	p. 155
Order Reduction	p. 155
Local Error Analysis	p. 158
Global Error Analysis	p. 161
Concluding Notes	p. 166
Linear Multistep Methods	p. 170
The Order Conditions	p. 171
Examples	p. 173
Stability Analysis	p. 174
Step Size Restrictions for Advection-Diffusion	p. 181
Convergence Analysis	p. 182
Monotone ODE Methods	p. 185
Linear Positivity for One-Step Methods	p. 185
Nonlinear Positivity for One-Step Methods	p. 189
Positivity for Multistep Methods	p. 192
Related Monotonicity Results	p. 196
Variable Step Size Control	p. 197
Step Size Selection	p. 197
An Explicit Runge-Kutta Example	p. 200
An Implicit Multistep Example	p. 203
General Purpose ODE Codes	p. 205
Numerical Examples	p. 206
A Model for Antibodies in Tumorous Tissue	p. 206
The Nonlinear Schrödinger Equation	p. 209
Advection-Diffusion Discretizations	p. 215
Non-oscillatory MOL Advection Discretizations	p. 215
Spatial Discretization for Linear Advection	p. 215
Numerical Examples	p. 222
Positivity and the TVD Property	p. 226
Nonlinear Scalar Conservation Laws	p. 233
Direct Space-Time Advection Discretizations	p. 239
Optimal-Order DST Schemes	p. 239
A Non-oscillatory Third-Order DST Scheme	p. 243
Explicit Schemes with Unconditional Stability	p. 248
Implicit Spatial Discretizations	p. 250
Order Conditions	p. 251
Examples	p. 253
Stability and Convergence	p. 258
Monotonicity	p. 261
Time Integration Aspects	p. 263
Non-uniform Grids - Finite Volumes (1D)	p. 264
Vertex Centered Schemes	p. 265
Cell Centered Schemes	p. 272
Numerical Illustrations	p. 278
Higher-Order Methods and Limiting	p. 281
Non-uniform Grids-Finite Elements(1D)	p. 283
The Basic Galerkin Method	p. 283
Standard Galerkin Error Estimates	p. 288
Upwinding	p. 291
Multi-dimensional Aspects	p. 292
Cartesian Grid Discretizations	p. 293
Diffusion on Cartesian Grids	p. 295
Advectionon Cartesian Grids	p. 303
Transformed Cartesian Grids	p. 308
Unstructured Grids	p. 311
Notes on Moving Grids and Grid Refinement	p. 316
Dynamic Regridding	p. 316
Static Regridding	p. 321
Splitting Methods	p. 325
Operator Splitting	p. 325
First-Order Splitting	p. 325
Second-Order Symmetrical Splitting	p. 329
Higher-Order Splittings	p. 330
Abstract Initial Value Problems	p. 331
Advection-Diffusion-Reaction Splittings	p. 335
Dimension Splitting	p. 337
Boundary Values and Stiff Terms	p. 344
LOD Methods	p. 348
The LOD-Backward Euler Method	p. 348
LOD Crank-Nicolson Methods	p. 351
The Trapezoidal Splitting Method	p. 359
Boundary Correction Techniques	p. 365
Numerical Comparisons	p. 367
ADI Methods	p. 369
The Peaceman-Rachford Method	p. 369
The Douglas Method	p. 373
IMEX Methods	p. 383
The IMEX- Method	p. 383
IMEX Multistep Methods	p. 386
Notes on IMEX Runge-Kutta Methods	p. 391
Concluding Remarks and Tests	p. 393
Rosenbrock AMF Methods	p. 398
One-Stage Methods of Order One and Two	p. 398
Two-Stage Methods of Order Two and Three	p. 400
A Three-Stage Method of Order Two	p. 403
Concluding Remarks and Tests	p. 405
Numerical Examples	p. 409
Two Chemo-taxis Problems from Biology	p. 409
The Numerical Methods	p. 411
Numerical Experiments	p. 412
Stabilized Explicit Runge-Kutta Methods	p. 419
The RKC Family	p. 420
Stability Polynomials	p. 420
Integration Formulas	p. 426
Internal Stability and Full Convergence Properties	p. 430
The ROCK Family	p. 433
Stability Polynomials	p. 433
Integration Formulas	p. 435
Internal Stability and Convergence	p. 436
Numerical Examples	p. 438
A Combustion Model	p. 439
A Radiation-Diffusion Model	p. 441
Bibliography	p. 447
Index	p. 465
Table of Contents provided by Publisher. All Rights Reserved.

Springer Computational Mathematics

At a Glance

Hardcover

Industry Reviews

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