Computer Simulation Using Particles

Transcription

1 Computer Simulation Using Particles R W Hockney Emeritus Professor, University ofreading and J W Eastwood Culham Laboratory, Abingdon Institute of Physics Publishing Bristol and Philadelphia

2 CONTENTS Foreword xv Preface to the Paperback Edition xvii Preface xix Chapter 1 Computer Experiments Using Particle Models Introduction The Computer Experiment The Role of the Computer Experiment Setting Up Computer Experiments Length and Time Scales Physical Systems Correlated Systems Uncorrelated (Collisionless) Systems Collisional Systems Collision-Dominated Systems Particle Models The Particle-Particle Method The Particle-Mesh Method The Particle-Particle-Particle-Mesh Method 22 Chapter 2 A One-Dimensionai Plasma Model The Physical System 24

3 VÜi CONTENTS 2-2 Discretization of the Mathematical Model The Superparticle Equations The Field Equations Charge Assignment and Force Interpolation The Discrete Model Numerical Algorithms Dimensionless Units Charge Assignment Poisson's Equation Computer Experiments The Two-Particle Test Wave Dispersion The Cold Plasma Energy Conservation The Two-Stream Instability 42 Chapter 3 The Simulation Program Introduction Program Requirements and Specification User's Requirements Program Specifications The OLYMPUS Programming System The Program ES 1D1V The Program Control Structure The Master Index Class 1: The Prologue Subprograms Calculation and Output Subprograms Final Remarks 92 Chapter 4 Time Integration Scheines Introduction Consistency Accuracy Stability The Root Locus Method The Amplification Matrix Efficiency The Leapfrog Harmonie Oscillator Examples of Integration Schemes Lorentz Force Integrators Viscous Force Integrators Low-Storage Runge-Kutta Schemes 117

4 CONTENTS ix Chapter 5 The Particle-Mesh Force Calculation Introduction Forces in One Dimension The Continuous System The NGP Scheme The CIC Scheme Mixed Schemes The Hierarchy of Charge Assignment Schemes The Long-Range Constraints The Smoothness Constraints The Momentum Conservation Constraint Cloud and Assignment Function Shapes Truncation Errors Energy-Conserving Schemes Transform Space Analysis Charge Assignment The Potential Solver Force Interpolation The Interparticle Force 164 Chapter 6 The Solution of the Field Equations Introduction Selection of Method Nonlinear Problems Newton Iteration Mesh Relaxation Jacobi Method (J) Gauss-Seidel (GS) Successive Overrelaxation (SOR) Chebyshev Acceleration Block Methods Alternating Direction Implicit (ADI) Matrix Methods Thomas Tridiagonal Algorithm Conjugate-Gradient Algorithm (CGA) Sparse Matrix Methods (SM) Incomplete Decomposition Stone's Strongly Implicit Procedure (SIP) Incomplete-Choleski-Conjugate-Gradient (ICCG) Rapid Elliptic Solvers (RES) Cyclic Reduction (CR) Multiple Fourier Transform (MFT) FACR Method 208

5 X CONTENTS Convolution Methods James'Algorithm Capacity Matrix Method Concus and Golub Iteration Concluding Remarks 221 Chapter 7 Collisionless Particle Models Introduction The Kinetic Equations Small-Timestep Limit Finite Timestep The Dispersion Relation Small-Timestep Limit Finite Timestep The Warm-Plasma Approximation Mode Coupling Finite Multidimensional Systems Periodicity Two and Three Dimensions Collisions Conservation Laws Energy Momentum Angular Momentum Optimization The Interparticle Force One-Dimensional Schemes Interlacing Force Averaging Harmonie Averaging Multidimensional Schemes 265 Chapter 8 Particle-Particle-Particle-Mesh (P 3 M) Algorithms Introduction Force Splitting The Mesh Force Charge Assignment The Force Calculation Errors in the Force The Short-Range Force The Chaining Mesh The Linked Lists The Momentum Change The Timing Equation 281

6 CONTENTS xi 8-6 Optimization Calculation of Force Accuracy Comparison of Scheines The Cost-Quality Relationship Practical Considerations The Program Data Organization Assignment and Interpolation The Potential Solver The Short-Range Force Parameter Selection 301 Chapter 9 Plasma Simulation Introduction Magnetohydrodynamics Electrostatic Plasma Historical Survey Two-Dimensional Electrostatic Model Collision Time HeatingTime Empirical Correlations Anomalous Diffusion Diffusion Experiment Supporting Theory Choice of Timestep and Mesh Size Criticism of the Experiment Two-and-a-Half- and Three-Dimensional Models Diagnostics and Display The Magnetosphere Magnetohydrodynamic Particle Model Overall Magnetosphere Ampere Particle Model Geomagnetic Tail 351 Chapter 10 Semiconductor Device Simulation Introduction Purpose of Simulation Defining the Problem Typesof Model Electron Flow in Semiconductors Equations of Motion Band Structure of Gallium Arsenide Scattering Processes Mobility Transient Relaxation Effects 372

7 XÜ CONTENTS 10-3 Designing the Computer Model Particle-Mesh Calculation Monte-Carlo Scattering Selection Modified Timestep Cycle Measurements on FETs Static Characteristics Looking Inside the FET Dynamic Characteristics Lumped-Parameter Equivalent Circuit Noise TheCOLDFET Complex Geometries 407 Chapter 11 Astrophysics Introduction Stellar Evolution The Gravitational N-Body Problem Collisional and Collisionless Systems Clustering of Stars and Galaxies The Big Bang Computer Simulation of Stellar Systems Small Clusters The Force Law Time Integration Spiral Galaxies Theories of Spiral Structure The Model Choice of Timestep and Mesh Size Collision Time and Particle Number The Ubiquitous Bar Instability Conditions for Spiral Structure The Protogalaxy Clustering of Galaxies Equations of the Expanding Universe Comoving Coordinates The Simulation Model Results and Conclusions 446 Chapter 12 Solids, Liquids, and Phase Changes Introduction Molecular Dynamics The Force Law Time Integration 459

8 CONTENTS Xlll 12-2 Two-Dimensional Electron Film Dimensionless Equations Choosing the Timestep Scaling the Problem Computer Time and Storage Melting the Electron Film Ionic Microcrystals Dimensionless Equations Choosing the Timestep Spatial Mesh and Computer Economy Thermodynamic Measurements Measurements in Different Regions Melting, Supercooling, and Glass Formation Radius Ratio, Hardness, and Size Effects Testing Theories of Melting 496 Appendix A Fourier Transforms, Fourier Series, and Finite Fourier Transforms 499 A-l Transform Definitions 499 A-l-1 The Fourier Transform (FT) 499 A-l-2 The Fourier Series (FS) 501 A-l-3 The Finite Fourier Transform (FFT) 501 A-2 Symmetries 502 A-3 Theorems 502 A-4 Special Functions 505 A-4-1 The "Top-Hat" Function U{x) 505 A-4-2 The "Triangle" Function A (x) 505 A-4-3 TheGaussian G(x) 505 A-4-4 The Dirac Delta Function ö(x) 505 A-4-5 The Sampling Function TII(x) 506 A-5 Relationship Between Transforms 507 A-6 Multidimensional Transforms 508 Bibliography 509 Index