Preface ...................................................... 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
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 examplel ............ 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 ...................................... 106
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 FluxLimiters ........................................ 114
6.12 TVDLimiters ......................................... 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 ............................. 181
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 Korm 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 ............................ 239
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 ............................... 335
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 ............................. 429
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 ... 509
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 ......................................................... 553
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