计算物理学 第2版 英文版PDF电子书下载

Part Ⅰ Numerical Methods 3

1 Error Analysis 3

1.1 Machine Numbers and Rounding Errors 3

1.2 Numerical Errors of Elementary Floating Point Operations 6

1.2.1 Numerical Extinction 7

1.2.2 Addition 8

1.2.3 Multiplication 9

1.3 Error Propagation 9

1.4 Stability of Iterative Algorithms 11

1.5 Example：Rotation 12

1.6 Truncation Error 13

1.7 Problems 14

2 Interpolation 15

2.1 Interpolating Functions 15

2.2 Polynomial Interpolation 16

2.2.1 Lagrange Polynomials 17

2.2.2 Barycentric Lagrange Interpolation 17

2.2.3 Newton's Divided Differences 18

2.2.4 Neville Method 20

2.2.5 Error of Polynomial Interpolation 21

2.3 Spline Interpolation 22

2.4 Rational Interpolation 25

2.4.1 PadéApproximant 25

2.4.2 Barycentric Rational Interpolation 27

2.5 Multivariate Interpolation 32

2.6 Problems 33

3 Numerical Differentiation 37

3.1 One-Sided Difference Quotient 37

3.2 Central Difference Quotient 38

3.3 Extrapolation Methods 39

3.4 Higher Derivatives 41

3.5 Partial Derivatives of Multivariate Functions 42

3.6 Problems 43

4 Numerical Integration 45

4.1 Equidistant Sample Points 46

4.1.1 Closed Newton-Cotes Formulae 46

4.1.2 Open Newton-Cotes Formulae 48

4.1.3 Composite Newton-Cotes Rules 48

4.1.4 Extrapolation Method（Romberg Integration） 49

4.2 Optimized Sample Points 50

4.2.1 Clenshaw-Curtis Expressions 50

4.2.2 Gaussian Integration 52

4.3 Problems 56

5 Systems of Inhomogeneous Linear Equations 59

5.1 Gaussian Elimination Method 60

5.1.1 Pivoting 63

5.1.2 Direct LU Decomposition 63

5.2 QR Decomposition 64

5.2.1 QR Decomposition by Orthogonalization 64

5.2.2 QR Decomposition by Householder Reflections 66

5.3 Linear Equations with Tridiagonal Matrix 69

5.4 Cyclic Tridiagonal Systems 71

5.5 Iterative Solution of Inhomogeneous Linear Equations 73

5.5.1 General Relaxation Method 73

5.5.2 Jacobi Method 73

5.5.3 Gauss-Seidel Method 74

5.5.4 Damping and Successive Over-Relaxation 75

5.6 Conjugate Gradients 76

5.7 Matrix Inversion 77

5.8 Problems 78

6 Roots and Extremal Points 83

6.1 Root Finding 83

6.1.1 Bisection 84

6.1.2 Regula Falsi（False Position）Method 85

6.1.3 Newton-Raphson Method 85

6.1.4 Secant Method 86

6.1.5 Interpolation 87

6.1.6 Inverse Interpolation 88

6.1.7 Combined Methods 91

6.1.8 Multidimensional Root Finding 97

6.1.9 Quasi-Newton Methods 98

6.2 Function Minimization 99

6.2.1 The Ternary Search Method 99

6.2.2 The Golden Section Search Method（Brent's Method） 101

6.2.3 Minimization in Multidimensions 106

6.2.4 Steepest Descent Method 106

6.2.5 Conjugate Gradient Method 107

6.2.6 Newton-Raphson Method 107

6.2.7 Quasi-Newton Methods 108

6.3 Problems 110

7 Fourier Transformation 113

7.1 Fourier Integral and Fourier Series 113

7.2 Discrete Fourier Transformation 114

7.2.1 Trigonometric Interpolation 116

7.2.2 Real Valued Functions 118

7.2.3 Approximate Continuous Fourier Transformation 119

7.3 Fourier Transform Algorithms 120

7.3.1 Goertzel's Algorithm 120

7.3.2 Fast Fourier Transformation 121

7.4 Problems 125

8 Random Numbers and Monte Carlo Methods 127

8.1 Some Basic Statistics 127

8.1.1 Probability Density and Cumulative Probability Distribution 127

8.1.2 Histogram 128

8.1.3 Expectation Values and Moments 129

8.1.4 Example：Fair Die 130

8.1.5 Normal Distribution 131

8.1.6 Multivariate Distributions 132

8.1.7 Central Limit Theorem 133

8.1.8 Example：Binomial Distribution 133

8.1.9 Average of Repeated Measurements 134

8.2 Random Numbers 135

8.2.1 Linear Congruent Mapping 135

8.2.2 Marsaglia-Zamann Method 135

8.2.3 Random Numbers with Given Distribution 136

8.2.4 Examples 136

8.3 Monte Carlo Integration 138

8.3.1 Numerical Calculation ofπ 138

8.3.2 Calculation of an Integral 139

8.3.3 More General Random Numbers 140

8.4 Monte Carlo Method for Thermodynamic Averages 141

8.4.1 Simple Sampling 141

8.4.2 Importance Sampling 142

8.4.3 Metropolis Algorithm 142

8.5 Problems 144

9 Eigenvalue Problems 147

9.1 Direct Solution 148

9.2 Jacobi Method 148

9.3 Tridiagonal Matrices 150

9.3.1 Characteristic Polynomial of a Tridiagonal Matrix 151

9.3.2 Special Tridiagonal Matrices 151

9.3.3 The QL Algorithm 156

9.4 Reduction to a Tridiagonal Matrix 157

9.5 Large Matrices 159

9.6 Problems 160

10 Data Fitting 161

10.1 Least Square Fit 162

10.1.1 Linear Least Square Fit 163

10.1.2 Linear Least Square Fit with Orthogonalization 165

10.2 Singular Value Decomposition 167

10.2.1 Full Singular Value Decomposition 168

10.2.2 Reduced Singular Value Decomposition 168

10.2.3 Low Rank Matrix Approximation 170

10.2.4 Linear Least Square Fit with Singular Value Decomposition 172

10.3 Problems 175

11 Discretization of Differential Equations 177

11.1 Classification of Differential Equations 178

11.1.1 Linear Second Order PDE 178

11.1.2 Conservation Laws 179

11.2 Finite Differences 180

11.2.1 Finite Differences in Time 181

11.2.2 Stability Analysis 182

11.2.3 Method of Lines 183

11.2.4 Eigenvector Expansion 183

11.3 Finite Volumes 185

11.3.1 Discretization of fluxes 188

11.4 Weighted Residual Based Methods 190

11.4.1 Point Collocation Method 191

11.4.2 Sub-domain Method 191

11.4.3 Least Squares Method 192

11.4.4 Galerkin Method 192

11.5 Spectral and Pseudo-spectral Methods 193

11.5.1 Fourier Pseudo-spectral Methods 193

11.5.2 Example：Polynomial Approximation 194

11.6 Finite Elements 196

11.6.1 One-Dimensional Elements 196

11.6.2 Two-and Three-Dimensional Elements 197

11.6.3 One-Dimensional Galerkin FEM 201

11.7 Boundary Element Method 204

12 Equations of Motion 207

12.1 The State Vector 208

12.2 Time Evolution of the State Vector 209

12.3 Explicit Forward Euler Method 210

12.4 Implicit Backward Euler Method 212

12.5 Improved Euler Methods 213

12.6 Taylor Series Methods 215

12.6.1 Nordsieck Predictor-Corrector Method 215

12.6.2 Gear Predictor-Corrector Methods 217

12.7 Runge-Kutta Methods 217

12.7.1 Second Order Runge-Kutta Method 218

12.7.2 Third Order Runge-Kutta Method 218

12.7.3 Fourth Order Runge-Kutta Method 219

12.8 Quality Control and Adaptive Step Size Control 220

12.9 Extrapolation Methods 221

12.10 Linear Multistep Methods 222

12.10.1 Adams-Bashforth Methods 222

12.10.2 Adams-Moulton Methods 223

12.10.3 Backward Differentiation（Gear）Methods 223

12.10.4 Predictor-Corrector Methods 224

12.11 Verlet Methods 225

12.11.1 Liouville Equation 225

12.11.2 Split-Operator Approximation 226

12.11.3 Position Verlet Method 227

12.11.4 Velocity Verlet Method 227

12.11.5 St？rmer-Verlet Method 228

12.11.6 Error Accumulation for the St？rmer-Verlet Method 229

12.11.7 Beeman's Method 230

12.11.8 The Leapfrog Method 231

12.12 Problems 232

Part Ⅱ Simulation of Classical and Quantum Systems 239

13 Rotational Motion 239

13.1 Transformation to a Body Fixed Coordinate System 239

13.2 Properties of the Rotation Matrix 240

13.3 Properties of W,Connection with the Vector of Angular Velocity 242

13.4 Transformation Properties of the Angular Velocity 244

13.5 Momentum and Angular Momentum 246

13.6 Equations of Motion of a Rigid Body 246

13.7 Moments of Inertia 247

13.8 Equations of Motion for a Rotor 248

13.9 Explicit Methods 248

13.10 Loss of Orthogonality 250

13.11 Implicit Method 251

13.12 Kinetic Energy of a Rotor 255

13.13 Parametrization by Euler Angles 255

13.14 Cayley-Klein Parameters,Quaternions,Euler Parameters 256

13.15 Solving the Equations of Motion with Quaternions 259

13.16 Problems 260

14 Molecular Mechanics 263

14.1 Atomic Coordinates 264

14.2 Force Fields 266

14.2.1 Intramolecular Forces 267

14.2.2 Intermolecular Interactions 269

14.3 Gradients 270

14.4 Normal Mode Analysis 274

14.4.1 Harmonic Approximation 274

14.5 Problems 276

15 Thermodynamic Systems 279

15.1 Simulation of a Lennard-Jones Fluid 279

15.1.1 Integration of the Equations of Motion 280

15.1.2 Boundary Conditions and Average Pressure 281

15.1.3 Initial Conditions and Average Temperature 281

15.1.4 Analysis of the Results 282

15.2 Monte Carlo Simulation 287

15.2.1 One-Dimensional Ising Model 287

15.2.2 Two-Dimensional Ising Model 289

15.3 Problems 290

16 Random Walk and Brownian Motion 293

16.1 Markovian Discrete Time Models 293

16.2 Random Walk in One Dimension 294

16.2.1 Random Walk with Constant Step Size 295

16.3 The Freely Jointed Chain 296

16.3.1 Basic Statistic Properties 297

16.3.2 Gyration Tensor 299

16.3.3 Hookean Spring Model 300

16.4 Langevin Dynamics 301

16.5 Problems 303

17 Electrostatics 305

17.1 Poisson Equation 305

17.1.1 Homogeneous Dielectric Medium 306

17.1.2 Numerical Methods for the Poisson Equation 307

17.1.3 Charged Sphere 309

17.1.4 Variable ε 311

17.1.5 Discontinuous ε 313

17.1.6 Solvation Energy of a Charged Sphere 314

17.1.7 The Shifted Grid Method 314

17.2 Poisson-Boltzmann Equation 315

17.2.1 Linearization of the Poisson-Boltzmann Equation 317

17.2.2 Discretization of the Linearized Poisson-Boltzmann Equation 318

17.3 Boundary Element Method for the Poisson Equation 318

17.3.1 Integral Equations for the Potential 318

17.3.2 Calculation of the Boundary Potential 321

17.4 Boundary Element Method for the Linearized Poisson-Boltzmann Equation 324

17.5 Electrostatic Interaction Energy（Onsager Model） 325

17.5.1 Example：Point Charge in a Spherical Cavity 326

17.6 Problems 327

18 Waves 329

18.1 Classical Waves 329

18.2 Spatial Discretization in One Dimension 332

18.3 Solution by an Eigenvector Expansion 334

18.4 Discretization of Space and Time 337

18.5 Numerical Integration with a Two-Step Method 338

18.6 Reduction to a First Order Differential Equation 340

18.7 Two-Variable Method 343

18.7.1 Leapfrog Scheme 343

18.7.2 Lax-Wendroff Scheme 345

18.7.3 Crank-Nicolson Scheme 347

18.8 Problems 349

19 Diffusion 351

19.1 Particle Flux and Concentration Changes 351

19.2 Diffusion in One Dimension 353

19.2.1 Explicit Euler（Forward Time Centered Space）Scheme 353

19.2.2 Implicit Euler（Backward Time Centered Space）Scheme 355

19.2.3 Crank-Nicolson Method 357

19.2.4 Error Order Analysis 358

19.2.5 Finite Element Discretization 360

19.3 Split-Operator Method for Multidimensions 360

19.4 Problems 362

20 Nonlinear Systems 363

20.1 Iterated Functions 364

20.1.1 Fixed Points and Stability 364

20.1.2 The Lyapunov Exponent 366

20.1.3 The Logistic Map 367

20.1.4 Fixed Points of the Logistic Map 367

20.1.5 Bifurcation Diagram 369

20.2 Population Dynamics 370

20.2.1 Equilibria and Stability 370

20.2.2 The Continuous Logistic Model 371

20.3 Lotka-Volterra Model 372

20.3.1 Stability Analysis 372

20.4 Functional Response 373

20.4.1 Holling-Tanner Model 375

20.5 Reaction-Diffusion Systems 378

20.5.1 General Properties of Reaction-Diffusion Systems 378

20.5.2 Chemical Reactions 378

20.5.3 Diffusive Population Dynamics 379

20.5.4 Stability Analysis 379

20.5.5 Lotka-Volterra Model with Diffusion 380

20.6 Problems 382

21 Simple Quantum Systems 385

21.1 Pure and Mixed Quantum States 386

21.1.1 Wavefunctions 387

21.1.2 Density Matrix for an Ensemble of Systems 387

21.1.3 Time Evolution of the Density Matrix 388

21.2 Wave Packet Morion in One Dimension 389

21.2.1 Discretization of the Kinetic Energy 390

21.2.2 Time Evolution 392

21.2.3 Example：Free Wave Packet Morion 402

21.3 Few-State Systems 403

21.3.1 Two-State System 405

21.3.2 Two-State System with Time Dependent Perturbation 408

21.3.3 Superexchange Model 410

21.3.4 Ladder Model for Exponential Decay 412

21.3.5 Landau-Zener Model 414

21.4 The Dissipative Two-State System 416

21.4.1 Equations of Morion for a Two-State System 416

21.4.2 The Vector Model 417

21.4.3 The Spin-1/2 System 418

21.4.4 Relaxation Processes—The Bloch Equations 420

21.4.5 The Driven Two-State System 421

21.4.6 Elementary Qubit Manipulation 428

21.5 Problems 430

Appendix Ⅰ Performing the Computer Experiments 433

Appendix Ⅱ Methods and Algorithms 435

References 441

Index 449