Modelling and Simulation in Plasma Physics for Physicists and Mathematicians

Preface xiii

Acknowledgements xvii

Preamble 1

Part I Continuum Methods 5

1 Foundations of Computational Fluid Mechanics 7

1.1 Basic Concepts of Fluid Mechanics 7

1.2 The Basic Equations of Fluid Mechanics 8

1.3 Ideal (Dissipationless) Flow -- Hyperbolic Equations 9

1.4 Formal Solution 11

1.5 Discontinuities 12

1.6 Plasma Fluid Dynamics 14

1.7 Basic Principles of Finite Differencing 15

1.8 Conservative Finite-Difference Approximations 19

1.9 Numerical Fluid Approximations 21

1.10 Grid Geometry 22

1.11 Control Volume Differencing 22

1.12 Mesh Types 24

2 Analytic and Quasi-Analytic Approximations 29

2.1 Analytic and Quasi-Analytic Methods 29

3 Numerical Fluid Dynamics in the Eulerian Scheme 41

3.1 An Introduction to Steady-State Engineering Design Models 41

3.2 Spatial Differencing 45

3.3 Generalised Euler Schemes 49

4 Lagrangian Systems 75

4.1 Lagrangian Fluid Dynamics 75

4.2 One-Dimensional von Neumann--Richtmyer Algorithm 76

4.3 Multidimensional Lagrangian Schemes 80

4.4 Choice of Method 88

5 Arbitrary Lagrangian--Eulerian Schemes 89

5.1 Introduction 89

5.2 Step 1: The Lagrangian Stage 91

5.3 Step 2: The Iteration Stage 91

5.4 Step 3: Mesh Generation 92

5.5 Step 4: Re-zoning 92

6 Hybrid or 11/2 d Schemes 95

6.1 Introduction 95

7 Magneto-Hydrodynamics 105

7.1 Introduction 105

7.2 The MHD Equations 106

7.3 MHD Equations 112

7.4 Conjugate Conservative Operators 114

7.5 Finite-Difference Approximations 115

7.6 Cylindrical Geometry -- Self-Generated Magnetic Fields 116

Part II Particle Methods 119

8 Particle in Cell Simulations 121

8.1 Introduction 121

Part III Stochastic Methods 135

9 Monte Carlo Schemes 137

9.1 Monte Carlo Methods 137

9.2 Monte Carlo Integration 143

9.3 RandomWalks 145

9.4 Nuclear Reactor Criticality 145

9.5 Thermodynamic Properties and Equation of State 151

10 Numerical Diffusion Schemes 167

10.1 Introduction 167

10.2 Split-Time-Step and ADI Methods for Solving Diffusion Problems in Orthogonal Cartesian Grid Systems 169

10.3 The Diffusion Matrix 174

11 Ion--Electron Equilibration 185

12 Ionisation--Recombination Models 187

12.1 Introduction 187

12.2 Collisional-Radiative Model 188

12.3 Two-Stage Model 202

References 247

Index 251