Cover image for Numerical time-dependent Partial differential equations for scientists and engineers
Numerical time-dependent Partial differential equations for scientists and engineers
Personal Author:
Mathematics in science and engineering, 213
Publication Information:
Amsterdam [Netherlands] ; Boston [i.e. Burlington, MA] : Academic Press, 2010
Physical Description:
x, 294 p. : ill. ; 24 cm.


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PSZ JB 30000010278004 QA377 B755 2010 Open Access Book Book

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It is the first text that in addition to standard convergence theory treats other necessary ingredients for successful numerical simulations of physical systems encountered by every practitioner. The book is aimed at users with interests ranging from application modeling to numerical analysis and scientific software development. It is strongly influenced by the authors research in in space physics, electrical and optical engineering, applied mathematics, numerical analysis and professional software development. The material is based on a year-long graduate course taught at the University of Arizona since 1989. The book covers the first two-semesters of a three semester series. The second semester is based on a semester-long project, while the third semester requirement consists of a particular methods course in specific disciplines like computational fluid dynamics, finite element method in mechanical engineering, computational physics, biology, chemistry, photonics, etc.

The first three chapters focus on basic properties of partial differential equations, including analysis of the dispersion relation, symmetries, particular solutions and instabilities of the PDEs; methods of discretization and convergence theory for initial value problems. The goal is to progress from observations of simple numerical artifacts like diffusion, damping, dispersion, and anisotropies to their analysis and management technique, as it is not always possible to completely eliminate them.

In the second part of the book we cover topics for which there are only sporadic theoretical results, while they are an integral part and often the most important part for successful numerical simulation. We adopt a more heuristic and practical approach using numerical methods of investigation and validation. The aim is teach students subtle key issues in order to separate physics from numerics. The following topics are addressed: Implementation of transparent and absorbing boundary conditions; Practical stability analysis in the presence of the boundaries and interfaces; Treatment of problems with different temporal/spatial scales either explicit or implicit; preservation of symmetries and additional constraints; physical regularization of singularities; resolution enhancement using adaptive mesh refinement and moving meshes.

Table of Contents

Prefacep. v
Contentsp. ix
1 Overview of Partial Differential Equationsp. 1
1.1 Examples of Partial Differential Equationsp. 1
1.2 Linearization and Dispersion Relationp. 7
1.3 Well-posedness, Regularity and the Solution Operatorp. 15
1.4 Physical Instabilitiesp. 20
1.5 Group Velocity, Wave Action and Wave Energy Equationsp. 45
1.6 Project Assignmentp. 52
1.7 Project Samplep. 53
2 Discretization Methodsp. 59
2.1 Polynomial Interpolation and Finite Differencesp. 60
2.2 Compact Finite Differences and Dispersion Preserving Schemesp. 74
2.3 Spectral Differentiationp. 79
2.4 Method of Weighted Residuals, Finite Element and Finite Volume Methodsp. 92
2.5 Project Assignmentp. 101
2.6 Project Samplep. 102
3 Convergence Theory for Initial Value Problemsp. 109
3.1 Introduction to Convergence Theoryp. 109
3.2 Lax-Richtmyer Equivalence Theoremp. 117
3.3 Von Neumann Analysis and Courant-Friedrichs-Levy Necessary Stability Conditionp. 130
3.4 Project Assignmentp. 139
3.5 Project Samplep. 140
4 Numerical Boundary Conditionsp. 145
4.1 Introduction to Numerical Boundary and Interface Conditionsp. 145
4.2 Transparent Boundary Conditions for Hyperbolic and Dispersive Systemsp. 147
4.3 Berenger's Perfectly Matched Layer Boundary Conditionsp. 155
4.4 Matrix Stability Analysis in the Presence of Boundaries and Interfacesp. 165
4.5 Project Samplep. 168
5 Problems with Multiple Temporal and Spatial Scalesp. 175
5.1 Examples of Weakly and Strongly Interacting Multiple Scalesp. 175
5.2 Stiff Ordinary Differential Equation Solversp. 187
5.3 Long-Time Integrators for Hamiltonian Systemsp. 190
5.4 Hyperbolic Conservation Lawsp. 210
5.5 Methods of Fractional Steps, Time-Split and Approximate Factorization Algorithmsp. 240
5.6 Project Samplep. 245
6 Numerical Grid Generationp. 251
6.1 Non-uniform Static Grids, Stability and Accuracy Issuesp. 251
6.2 Adaptive and Moving Grids Based on Equidistribution Principlep. 256
6.3 Level Set Methodsp. 258
6.4 The Front Tracking Methodp. 263
6.5 Project Samplep. 268
Bibliographyp. 273
Indexp. 289