كتاب Finite Volume Methods for Hyperbolic Problems
منتدى هندسة الإنتاج والتصميم الميكانيكى
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 كتاب Finite Volume Methods for Hyperbolic Problems

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مُساهمةموضوع: كتاب Finite Volume Methods for Hyperbolic Problems   كتاب Finite Volume Methods for Hyperbolic Problems Emptyالجمعة 20 أبريل 2012, 9:09 pm

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Finite Volume Methods for Hyperbolic Problems
RANDALL J. LEVEQUE
University of Washington

كتاب Finite Volume Methods for Hyperbolic Problems F_v_m_10
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Contents
Preface page xvii
1 Introduction 1
1.1 Conservation Laws 3
1.2 Finite Volume Methods 5
1.3 Multidimensional Problems 6
1.4 Linear Waves and Discontinuous Media 7
1.5 CLAWPACK Software 8
1.6 References 9
1.7 Notation 10
Part I Linear Equations
2 Conservation Laws and Differential Equations 15
2.1 The Advection Equation 17
2.2 Diffusion and the Advection–Diffusion Equation 20
2.3 The Heat Equation 21
2.4 Capacity Functions 22
2.5 Source Terms 22
2.6 Nonlinear Equations in Fluid Dynamics 23
2.7 Linear Acoustics 26
2.8 Sound Waves 29
2.9 Hyperbolicity of Linear Systems 31
2.10 Variable-Coefficient Hyperbolic Systems 33
2.11 Hyperbolicity of Quasilinear and Nonlinear Systems 34
2.12 Solid Mechanics and Elastic Waves 35
2.13 Lagrangian Gas Dynamics and the p-System 41
2.14 Electromagnetic Waves 43
Exercises 46
3 Characteristics and Riemann Problems for Linear
Hyperbolic Equations 47
3.1 Solution to the Cauchy Problem 47
ixx Contents
3.2 Superposition of Waves and Characteristic Variables 48
3.3 Left Eigenvectors 49
3.4 Simple Waves 49
3.5 Acoustics 49
3.6 Domain of Dependence and Range of Influence 50
3.7 Discontinuous Solutions 52
3.8 The Riemann Problem for a Linear System 52
3.9 The Phase Plane for Systems of Two Equations 55
3.10 Coupled Acoustics and Advection 57
3.11 Initial–Boundary-Value Problems 59
Exercises 62
4 Finite Volume Methods 64
4.1 General Formulation for Conservation Laws 64
4.2 A Numerical Flux for the Diffusion Equation 66
4.3 Necessary Components for Convergence 67
4.4 The CFL Condition 68
4.5 An Unstable Flux 71
4.6 The Lax–Friedrichs Method 71
4.7 The Richtmyer Two-Step Lax–Wendroff Method 72
4.8 Upwind Methods 72
4.9 The Upwind Method for Advection 73
4.10 Godunov’s Method for Linear Systems 76
4.11 The Numerical Flux Function for Godunov’s Method 78
4.12 The Wave-Propagation Form of Godunov’s Method 78
4.13 Flux-Difference vs. Flux-Vector Splitting 83
4.14 Roe’s Method 84
Exercises 85
5 Introduction to the CLAWPACK Software 87
5.1 Basic Framework 87
5.2 Obtaining CLAWPACK 89
5.3 Getting Started 89
5.4 Using CLAWPACK – a Guide through example1 91
5.5 Other User-Supplied Routines and Files 98
5.6 Auxiliary Arrays and setaux.f 98
5.7 An Acoustics Example 99
Exercises 99
6 High-Resolution Methods 100
6.1 The Lax–Wendroff Method 100
6.2 The Beam–Warming Method 102
6.3 Preview of Limiters 103
6.4 The REA Algorithm with Piecewise Linear Reconstruction 106Contents xi
6.5 Choice of Slopes 107
6.6 Oscillations 108
6.7 Total Variation 109
6.8 TVD Methods Based on the REA Algorithm 110
6.9 Slope-Limiter Methods 111
6.10 Flux Formulation with Piecewise Linear Reconstruction 112
6.11 Flux Limiters 114
6.12 TVD Limiters 115
6.13 High-Resolution Methods for Systems 118
6.14 Implementation 120
6.15 Extension to Nonlinear Systems 121
6.16 Capacity-Form Differencing 122
6.17 Nonuniform Grids 123
Exercises 127
7 Boundary Conditions and Ghost Cells 129
7.1 Periodic Boundary Conditions 130
7.2 Advection 130
7.3 Acoustics 133
Exercises 138
8 Convergence, Accuracy, and Stability 139
8.1 Convergence 139
8.2 One-Step and Local Truncation Errors 141
8.3 Stability Theory 143
8.4 Accuracy at Extrema 149
8.5 Order of Accuracy Isn’t Everything 150
8.6 Modified Equations 151
8.7 Accuracy Near Discontinuities 155
Exercises 156
9 Variable-Coefficient Linear Equations 158
9.1 Advection in a Pipe 159
9.2 Finite Volume Methods 161
9.3 The Color Equation 162
9.4 The Conservative Advection Equation 164
9.5 Edge Velocities 169
9.6 Variable-Coefficient Acoustics Equations 171
9.7 Constant-Impedance Media 172
9.8 Variable Impedance 173
9.9 Solving the Riemann Problem for Acoustics 177
9.10 Transmission and Reflection Coefficients 178
9.11 Godunov’s Method 179
9.12 High-Resolution Methods 181xii Contents
9.13 Wave Limiters 181
9.14 Homogenization of Rapidly Varying Coefficients 183
Exercises 187
10 Other Approaches to High Resolution 188
10.1 Centered-in-Time Fluxes 188
10.2 Higher-Order High-Resolution Methods 190
10.3 Limitations of the Lax–Wendroff (Taylor Series) Approach 191
10.4 Semidiscrete Methods plus Time Stepping 191
10.5 Staggered Grids and Central Schemes 198
Exercises 200
Part II Nonlinear Equations
11 Nonlinear Scalar Conservation Laws 203
11.1 Traffic Flow 203
11.2 Quasilinear Form and Characteristics 206
11.3 Burgers’ Equation 208
11.4 Rarefaction Waves 209
11.5 Compression Waves 210
11.6 Vanishing Viscosity 210
11.7 Equal-Area Rule 211
11.8 Shock Speed 212
11.9 The Rankine–Hugoniot Conditions for Systems 213
11.10 Similarity Solutions and Centered Rarefactions 214
11.11 Weak Solutions 215
11.12 Manipulating Conservation Laws 216
11.13 Nonuniqueness, Admissibility, and Entropy Conditions 216
11.14 Entropy Functions 219
11.15 Long-Time Behavior and N-Wave Decay 222
Exercises 224
12 Finite Volume Methods for Nonlinear Scalar
Conservation Laws 227
12.1 Godunov’s Method 227
12.2 Fluctuations, Waves, and Speeds 229
12.3 Transonic Rarefactions and an Entropy Fix 230
12.4 Numerical Viscosity 232
12.5 The Lax–Friedrichs and Local Lax–Friedrichs Methods 232
12.6 The Engquist–Osher Method 234
12.7 E-schemes 235
12.8 High-Resolution TVD Methods 235
12.9 The Importance of Conservation Form 237
12.10 The Lax–Wendroff Theorem 239Contents xiii
12.11 The Entropy Condition 243
12.12 Nonlinear Stability 244
Exercises 252
13 Nonlinear Systems of Conservation Laws 253
13.1 The Shallow Water Equations 254
13.2 Dam-Break and Riemann Problems 259
13.3 Characteristic Structure 260
13.4 A Two-Shock Riemann Solution 262
13.5 Weak Waves and the Linearized Problem 263
13.6 Strategy for Solving the Riemann Problem 263
13.7 Shock Waves and Hugoniot Loci 264
13.8 Simple Waves and Rarefactions 269
13.9 Solving the Dam-Break Problem 279
13.10 The General Riemann Solver for Shallow Water Equations 281
13.11 Shock Collision Problems 282
13.12 Linear Degeneracy and Contact Discontinuities 283
Exercises 287
14 Gas Dynamics and the Euler Equations 291
14.1 Pressure 291
14.2 Energy 292
14.3 The Euler Equations 293
14.4 Polytropic Ideal Gas 293
14.5 Entropy 295
14.6 Isothermal Flow 298
14.7 The Euler Equations in Primitive Variables 298
14.8 The Riemann Problem for the Euler Equations 300
14.9 Contact Discontinuities 301
14.10 Riemann Invariants 302
14.11 Solution to the Riemann Problem 302
14.12 The Structure of Rarefaction Waves 305
14.13 Shock Tubes and Riemann Problems 306
14.14 Multifluid Problems 308
14.15 Other Equations of State and Incompressible Flow 309
15 Finite Volume Methods for Nonlinear Systems 311
15.1 Godunov’s Method 311
15.2 Convergence of Godunov’s Method 313
15.3 Approximate Riemann Solvers 314
15.4 High-Resolution Methods for Nonlinear Systems 329
15.5 An Alternative Wave-Propagation Implementation of Approximate
Riemann Solvers 333
15.6 Second-Order Accuracy 335xiv Contents
15.7 Flux-Vector Splitting 338
15.8 Total Variation for Systems of Equations 340
Exercises 348
16 Some Nonclassical Hyperbolic Problems 350
16.1 Nonconvex Flux Functions 350
16.2 Nonstrictly Hyperbolic Problems 358
16.3 Loss of Hyperbolicity 362
16.4 Spatially Varying Flux Functions 368
16.5 Nonconservative Nonlinear Hyperbolic Equations 371
16.6 Nonconservative Transport Equations 372
Exercises 374
17 Source Terms and Balance Laws 375
17.1 Fractional-Step Methods 377
17.2 An Advection–Reaction Equation 378
17.3 General Formulation of Fractional-Step Methods for Linear Problems 384
17.4 Strang Splitting 387
17.5 Accuracy of Godunov and Strang Splittings 388
17.6 Choice of ODE Solver 389
17.7 Implicit Methods, Viscous Terms, and Higher-Order Derivatives 390
17.8 Steady-State Solutions 391
17.9 Boundary Conditions for Fractional-Step Methods 393
17.10 Stiff and Singular Source Terms 396
17.11 Linear Traffic Flow with On-Ramps or Exits 396
17.12 Rankine–Hugoniot Jump Conditions at a Singular Source 397
17.13 Nonlinear Traffic Flow with On-Ramps or Exits 398
17.14 Accurate Solution of Quasisteady Problems 399
17.15 Burgers Equation with a Stiff Source Term 401
17.16 Numerical Difficulties with Stiff Source Terms 404
17.17 Relaxation Systems 410
17.18 Relaxation Schemes 415
Exercises 416
Part III Multidimensional Problems
18 Multidimensional Hyperbolic Problems 421
18.1 Derivation of Conservation Laws 421
18.2 Advection 423
18.3 Compressible Flow 424
18.4 Acoustics 425
18.5 Hyperbolicity 425
18.6 Three-Dimensional Systems 428
18.7 Shallow Water Equations 429Contents xv
18.8 Euler Equations 431
18.9 Symmetry and Reduction of Dimension 433
Exercises 434
19 Multidimensional Numerical Methods 436
19.1 Finite Difference Methods 436
19.2 Finite Volume Methods and Approaches to Discretization 438
19.3 Fully Discrete Flux-Differencing Methods 439
19.4 Semidiscrete Methods with Runge–Kutta Time Stepping 443
19.5 Dimensional Splitting 444
Exercise 446
20 Multidimensional Scalar Equations 447
20.1 The Donor-Cell Upwind Method for Advection 447
20.2 The Corner-Transport Upwind Method for Advection 449
20.3 Wave-Propagation Implementation of the CTU Method 450
20.4 von Neumann Stability Analysis 452
20.5 The CTU Method for Variable-Coefficient Advection 453
20.6 High-Resolution Correction Terms 456
20.7 Relation to the Lax–Wendroff Method 456
20.8 Divergence-Free Velocity Fields 457
20.9 Nonlinear Scalar Conservation Laws 460
20.10 Convergence 464
Exercises 467
21 Multidimensional Systems 469
21.1 Constant-Coefficient Linear Systems 469
21.2 The Wave-Propagation Approach to Accumulating Fluxes 471
21.3 CLAWPACK Implementation 473
21.4 Acoustics 474
21.5 Acoustics in Heterogeneous Media 476
21.6 Transverse Riemann Solvers for Nonlinear Systems 480
21.7 Shallow Water Equations 480
21.8 Boundary Conditions 485
22 Elastic Waves 491
22.1 Derivation of the Elasticity Equations 492
22.2 The Plane-Strain Equations of Two-Dimensional Elasticity 499
22.3 One-Dimensional Slices 502
22.4 Boundary Conditions 502
22.5 The Plane-Stress Equations and Two-Dimensional Plates 504
22.6 A One-Dimensional Rod 509
22.7 Two-Dimensional Elasticity in Heterogeneous Media 509xvi Contents
23 Finite Volume Methods on Quadrilateral Grids 514
23.1 Cell Averages and Interface Fluxes 515
23.2 Logically Rectangular Grids 517
23.3 Godunov’s Method 518
23.4 Fluctuation Form 519
23.5 Advection Equations 520
23.6 Acoustics 525
23.7 Shallow Water and Euler Equations 530
23.8 Using CLAWPACK on Quadrilateral Grids 531
23.9 Boundary Conditions 534
Bibliography 535
Index 55


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