Finite Volume Methods for Hyperbolic Problems

Dedication......Page 9
Contents......Page 11
Preface......Page 19
1 Introduction......Page 23
1.1 Conservation Laws......Page 25
1.1.2 Discontinuous Solutions......Page 26
1.2.1 Riemann Problems......Page 27
1.3 Multidimensional Problems......Page 28
1.4 Linear Waves and Discontinuous Media......Page 29
1.5 CLAWPACK Software......Page 30
1.6 References......Page 31
1.7 Notation......Page 32
Part one Linear Equations......Page 35
2 Conservation Laws and Differential Equations......Page 37
2.1 The Advection Equation......Page 39
2.1.1 Variable Coefficients......Page 41
2.2 Diffusion and the Advection–Diffusion Equation......Page 42
2.3 The Heat Equation......Page 43
2.5.1 External Heat Sources......Page 44
2.6 Nonlinear Equations in Fluid Dynamics......Page 45
2.7 Linear Acoustics......Page 48
2.8 Sound Waves......Page 51
2.9 Hyperbolicity of Linear Systems......Page 53
2.10 Variable-Coefficient Hyperbolic Systems......Page 55
2.11 Hyperbolicity of Quasilinear and Nonlinear Systems......Page 56
2.12 Solid Mechanics and Elastic Waves......Page 57
2.12.1 Elastic Deformations......Page 58
2.12.2 Strain......Page 59
2.12.3 Stress......Page 60
2.12.4 The Equations of Motion......Page 61
2.13 Lagrangian Gas Dynamics and the Rho-System......Page 63
2.14 Electromagnetic Waves......Page 65
Exercises......Page 68
3.1 Solution to the Cauchy Problem......Page 69
3.2 Superposition of Waves and Characteristic Variables......Page 70
3.5 Acoustics......Page 71
3.6 Domain of Dependence and Range of Influence......Page 72
3.8 The Riemann Problem for a Linear System......Page 74
3.9 The Phase Plane for Systems of Two Equations......Page 77
3.9.1 Acoustics......Page 78
3.10 Coupled Acoustics and Advection......Page 79
3.11 Initial–Boundary-Value Problems......Page 81
Exercises......Page 84
4.1 General Formulation for Conservation Laws......Page 86
4.2 A Numerical Flux for the Diffusion Equation......Page 88
4.3 Necessary Components for Convergence......Page 89
4.4 The CFL Condition......Page 90
4.6 The Lax–Friedrichs Method......Page 93
4.8 Upwind Methods......Page 94
4.9 The Upwind Method for Advection......Page 95
Algorithm 4.1 (REA).......Page 98
4.12 The Wave-Propagation Form of Godunov’s Method......Page 100
4.13 Flux-Difference vs. Flux-Vector Splitting......Page 105
4.14 Roe’s Method......Page 106
Exercises......Page 107
5.1 Basic Framework......Page 109
5.3 Getting Started......Page 111
5.3.4 Makefiles......Page 112
5.4.1 The Main Program (driver.f )......Page 113
5.4.3 The claw1ez Routine......Page 114
5.4.5 The Riemann Solver......Page 115
5.4.6 The Input File claw1ez.data......Page 116
5.6 Auxiliary Arrays and setaux.f......Page 120
Exercises......Page 121
6.1 The Lax–Wendroff Method......Page 122
6.2 The Beam–Warming Method......Page 124
6.3 Preview of Limiters......Page 125
6.4 The REA Algorithm with Piecewise Linear Reconstruction......Page 128
6.5 Choice of Slopes......Page 129
6.6 Oscillations......Page 130
6.7 Total Variation......Page 131
6.8 TVD Methods Based on the REA Algorithm......Page 132
6.9 Slope-Limiter Methods......Page 133
6.10 Flux Formulation with Piecewise Linear Reconstruction......Page 134
6.11 Flux Limiters......Page 136
6.12 TVD Limiters......Page 137
6.13 High-Resolution Methods for Systems......Page 140
6.14 Implementation......Page 142
6.15 Extension to Nonlinear Systems......Page 143
6.16 Capacity-Form Differencing......Page 144
6.17 Nonuniform Grids......Page 145
6.17.1 High-Resolution Corrections......Page 148
Exercises......Page 149
7 Boundary Conditions and Ghost Cells......Page 151
7.2.1 Outflow Boundaries......Page 152
7.2.2 Inflow Boundary Conditions......Page 154
7.3 Acoustics......Page 155
7.3.1 Nonreflecting Boundary Conditions......Page 156
7.3.2 Incoming Waves......Page 157
7.3.3 Solid Walls......Page 158
7.3.4 Oscillating Walls......Page 159
Exercises......Page 160
8.1 Convergence......Page 161
8.1.1 Choice of Norms......Page 162
8.2 One-Step and Local Truncation Errors......Page 163
8.3 Stability Theory......Page 165
8.3.1 Contractive Operators......Page 166
8.3.2 Lax–Richtmyer Stability for Linear Methods......Page 167
8.3.3 2-Norm Stability and von Neumann Analysis......Page 168
8.3.4 1-Norm Stability of the Upwind Method......Page 169
8.3.5 Total-Variation Stability for Nonlinear Methods......Page 170
8.4 Accuracy at Extrema......Page 171
8.5 Order of Accuracy Isn’t Everything......Page 172
8.6 Modified Equations......Page 173
8.6.1 The Upwind Method......Page 174
8.6.2 Lax–Wendroff Method......Page 176
8.7 Accuracy Near Discontinuities......Page 177
Exercises......Page 178
9 Variable-Coefficient Linear Equations......Page 180
9.1 Advection in a Pipe......Page 181
9.2 Finite Volume Methods......Page 183
9.3 The Color Equation......Page 184
9.3.1 High-Resolution Corrections......Page 185
9.4 The Conservative Advection Equation......Page 186
9.4.1 Conveyer Belts......Page 188
9.4.2 Traffic Flow......Page 189
9.5.1 The Color Equation......Page 191
9.5.2 The Conservative Equation......Page 192
9.6 Variable-Coefficient Acoustics Equations......Page 193
9.7 Constant-Impedance Media......Page 194
9.8 Variable Impedance......Page 195
9.9 Solving the Riemann Problem for Acoustics......Page 199
9.10 Transmission and Reflection Coefficients......Page 200
9.11 Godunov’s Method......Page 201
9.13 Wave Limiters......Page 203
9.14 Homogenization of Rapidly Varying Coefficients......Page 205
Exercises......Page 209
10.1 Centered-in-Time Fluxes......Page 210
10.2 Higher-Order High-Resolution Methods......Page 212
10.4 Semidiscrete Methods plus Time Stepping......Page 213
10.4.1 Evolution Equations for the Cell Averages......Page 214
10.4.2 TVD Time Stepping......Page 216
10.4.3 Reconstruction by Primitive Functions......Page 217
10.4.4 ENO Methods......Page 219
10.5 Staggered Grids and Central Schemes......Page 220
Exercises......Page 222
Part two Nonlinear Equations......Page 223
11.1 Traffic Flow......Page 225
11.2 Quasilinear Form and Characteristics......Page 228
11.3 Burgers’ Equation......Page 230
11.4 Rarefaction Waves......Page 231
11.6 Vanishing Viscosity......Page 232
11.7 Equal-Area Rule......Page 233
11.8 Shock Speed......Page 234
11.9 The Rankine–Hugoniot Conditions for Systems......Page 235
11.10 Similarity Solutions and Centered Rarefactions......Page 236
11.11 Weak Solutions......Page 237
11.12 Manipulating Conservation Laws......Page 238
11.14 Entropy Functions......Page 241
11.15 Long-Time Behavior and N-Wave Decay......Page 244
Exercises......Page 246
12.1 Godunov’s Method......Page 249
12.2 Fluctuations, Waves, and Speeds......Page 251
12.3 Transonic Rarefactions and an Entropy Fix......Page 252
12.5 The Lax–Friedrichs and Local Lax–Friedrichs Methods......Page 254
12.6 The Engquist–Osher method......Page 256
12.8 High-Resolution TVD Methods......Page 257
12.9 The Importance of Conservation Form......Page 259
12.10 The Lax–Wendroff Theorem......Page 261
12.11.1 Entropy Consistency of Godunov’s Method......Page 265
12.12 Nonlinear Stability......Page 266
12.12.1 Convergence Notions......Page 267
12.12.2 Compactness......Page 268
12.12.3 Function Spaces......Page 270
12.12.4 Total-Variation Stability......Page 271
Exercises......Page 274
13 Nonlinear Systems of Conservation Laws......Page 275
13.1 The Shallow Water Equations......Page 276
13.2 Dam-Break and Riemann Problems......Page 281
13.3 Characteristic Structure......Page 282
13.4 A Two-Shock Riemann Solution......Page 284
13.6 Strategy for Solving the Riemann Problem......Page 285
13.7 Shock Waves and Hugoniot Loci......Page 286
13.7.1 The All-Shock Riemann Solution......Page 288
13.7.2 The Entropy Condition......Page 289
13.8 Simple Waves and Rarefactions......Page 291
13.8.1 Integral Curves......Page 292
13.8.2 Riemann Invariants......Page 294
13.8.3 Simple Waves......Page 295
13.8.4 Genuine Nonlinearity and Linear Degeneracy......Page 296
13.8.5 Centered Rarefaction Waves......Page 297
13.8.6 The All-Rarefaction Riemann Solution......Page 299
13.9 Solving the Dam-Break Problem......Page 301
13.10 The General Riemann Solver for Shallow Water Equations......Page 303
13.11 Shock Collision Problems......Page 304
13.12 Linear Degeneracy and Contact Discontinuities......Page 305
13.12.1 Shallow Water Equations with a Passive Tracer......Page 306
13.12.2 The Riemann Problem and Contact Discontinuities......Page 308
Exercises......Page 309
14.1 Pressure......Page 313
14.2 Energy......Page 314
14.4 Polytropic Ideal Gas......Page 315
14.5 Entropy......Page 317
14.5.1 Isentropic Flow......Page 318
14.7 The Euler Equations in Primitive Variables......Page 320
14.8 The Riemann Problem for the Euler Equations......Page 322
14.9 Contact Discontinuities......Page 323
14.11 Solution to the Riemann Problem......Page 324
14.12 The Structure of Rarefaction Waves......Page 327
14.13 Shock Tubes and Riemann Problems......Page 328
14.14 Multifluid Problems......Page 330
14.15 Other Equations of State and Incompressible Flow......Page 331
15.1 Godunov’s Method......Page 333
15.2 Convergence of Godunov’s Method......Page 335
15.3 Approximate Riemann Solvers......Page 336
15.3.1 Linearized Riemann Solvers......Page 337
15.3.2 Roe Linearization......Page 339
15.3.3 Roe Solver for the Shallow Water Equations......Page 342
15.3.4 Roe Solver for the Euler Equations......Page 344
15.3.5 Sonic Entropy Fixes......Page 345
The Harten–Hyman Entropy Fix......Page 346
Numerical Viscosity......Page 347
The LLF Entropy Fix......Page 348
15.3.6 Failure of Linearized Solvers......Page 349
15.3.7 The HLL and HLLE Solvers......Page 350
15.4 High-Resolution Methods for Nonlinear Systems......Page 351
15.5 An Alternative Wave-Propagation Implementation of Approximate Riemann Solvers......Page 355
15.6 Second-Order Accuracy......Page 357
15.6.1 Two-Step Lax–Wendroff Methods......Page 359
15.7.1 The Steger–Warming flux......Page 360
15.8 Total Variation for Systems of Equations......Page 362
15.8.1 Total Variation Estimates for Linear Systems......Page 363
15.8.2 Wave-Strength Estimates for Nonlinear Systems......Page 366
15.8.3 Glimm’s Random-Choice Method......Page 367
Start-up Errors......Page 368
Slow-Moving Shocks......Page 369
Exercises......Page 370
16.1 Nonconvex Flux Functions......Page 372
16.1.1 Two-Phase Flow and the Buckley–Leverett Equation......Page 373
16.1.2 Solving Nonconvex Riemann Problems......Page 375
Convex-Hull Construction......Page 376
Osher’s Solution......Page 377
16.1.3 Finite Volume Methods for Nonconvex Problems......Page 378
16.2.1 Uncoupled Advection Equations......Page 380
16.2.2 Uncoupled Burgers’ Equations......Page 382
16.2.3 Undercompressive and Overcompressive Shocks......Page 383
16.3.1 A Weakly-hyperbolic System......Page 384
Singular-Source Interpretation......Page 386
Physical Interpretation......Page 387
16.3.2 Equations of Mixed Type......Page 388
16.4 Spatially Varying Flux Functions......Page 390
16.4.1 Traffic Flow with a Varying Speed Limit......Page 391
16.4.2 Rewriting the Equations as an Autonomous System......Page 392
16.5 Nonconservative Nonlinear Hyperbolic Equations......Page 393
16.6 Nonconservative Transport Equations......Page 394
Exercises......Page 396
External Forces Such as Gravity......Page 397
Higher-Order Derivatives......Page 398
17.1 Fractional-Step Methods......Page 399
17.2.1 An Unsplit Method......Page 400
17.2.2 A Fractional-Step Method......Page 402
17.3 General Formulation of Fractional-Step Methods for Linear Problems......Page 406
17.4 Strang Splitting......Page 409
17.5 Accuracy of Godunov and Strang Splittings......Page 410
17.6 Choice of ODE Solver......Page 411
17.7 Implicit Methods, Viscous Terms, and Higher-Order Derivatives......Page 412
17.8 Steady-State Solutions......Page 413
17.9 Boundary Conditions for Fractional-Step Methods......Page 415
17.11 Linear Traffic Flow with On-Ramps or Exits......Page 418
17.12 Rankine–Hugoniot Jump Conditions at a Singular Source......Page 419
17.13 Nonlinear Traffic Flow with On-Ramps or Exits......Page 420
17.14 Accurate Solution of Quasisteady Problems......Page 421
17.15 Burgers Equation with a Stiff Source Term......Page 423
17.16 Numerical Difficulties with Stiff Source Terms......Page 426
17.17 Relaxation Systems......Page 432
17.17.1 Chapman–Enskog Expansion......Page 434
17.17.3 Thermal Relaxation and Isothermal Flow......Page 435
17.18 Relaxation Schemes......Page 437
Exercises......Page 438
Part three Multidimensional Problems......Page 441
18.1 Derivation of Conservation Laws......Page 443
18.2 Advection......Page 445
18.3 Compressible Flow......Page 446
18.5 Hyperbolicity......Page 447
18.6 Three-Dimensional Systems......Page 450
18.7 Shallow Water Equations......Page 451
18.7.1 The Plane-Wave Riemann Problem......Page 452
18.8 Euler Equations......Page 453
18.8.2 Three-Dimensional Euler Equations......Page 454
18.9 Symmetry and Reduction of Dimension......Page 455
Exercises......Page 456
19.1.1 Taylor Series Expansion of the Exact Solution......Page 458
19.1.2 The Lax–Wendroff Method......Page 459
19.2 Finite Volume Methods and Approaches to Discretization......Page 460
19.3 Fully Discrete Flux-Differencing Methods......Page 461
19.3.2 Godunov’s Method......Page 463
19.3.3 Fluctuation Form......Page 464
19.4 Semidiscrete Methods with Runge–Kutta Time Stepping......Page 465
19.5 Dimensional Splitting......Page 466
19.5.1 CLAWPACK Implementation......Page 467
Exercise......Page 468
20.1 The Donor-Cell Upwind Method for Advection......Page 469
20.2 The Corner-Transport Upwind Method for Advection......Page 471
20.3 Wave-Propagation Implementation of the CTU Method......Page 472
20.4 von Neumann Stability Analysis......Page 474
20.5 The CTU Method for Variable-Coefficient Advection......Page 475
20.7 Relation to the Lax–Wendroff Method......Page 478
20.8 Divergence-Free Velocity Fields......Page 479
20.8.1 Stream-Function Specification of Velocities......Page 481
20.9 Nonlinear Scalar Conservation Laws......Page 482
20.10.1 Convergence of Dimensional Splitting......Page 486
20.10.2 Total Variation in Two Dimensions......Page 488
Exercises......Page 489
21.1 Constant-Coefficient Linear Systems......Page 491
21.2 The Wave-Propagation Approach to Accumulating Fluxes......Page 493
21.3 CLAWPACK Implementation......Page 495
21.4 Acoustics......Page 496
21.5 Acoustics in Heterogeneous Media......Page 498
21.5.1 Transverse Propagation......Page 499
21.7 Shallow Water Equations......Page 502
21.7.1 A Radial Dam-Break Problem......Page 504
21.8 Boundary Conditions......Page 507
21.8.3 Solid Walls......Page 508
21.8.4 No-Slip Boundary Condition......Page 509
21.8.5 Extrapolation and Absorbing Boundary Conditions......Page 510
22 Elastic Waves......Page 513
22.1 Derivation of the Elasticity Equations......Page 514
22.1.1 Linear Elasticity......Page 515
22.1.2 The Bulk Modulus and Acoustics......Page 520
22.2 The Plane-Strain Equations of Two-Dimensional Elasticity......Page 521
22.4 Boundary Conditions......Page 524
22.4.2 Specified Traction......Page 525
22.5 The Plane-Stress Equations and Two-Dimensional Plates......Page 526
22.7 Two-Dimensional Elasticity in Heterogeneous Media......Page 531
23 Finite Volume Methods on Quadrilateral Grids......Page 536
23.1 Cell Averages and Interface Fluxes......Page 537
23.2 Logically Rectangular Grids......Page 539
23.3 Godunov’s Method......Page 540
23.4 Fluctuation Form......Page 541
23.5 Advection Equations......Page 542
23.5.1 Fluctuation Form......Page 543
23.5.2 Variable-Coefficient Advection......Page 544
23.6.1 Solving the Normal Riemann Problem......Page 547
23.6.2 Transverse Riemann Solvers......Page 549
23.6.3 Solving the Riemann Problem by Rotating the Data......Page 550
23.7 Shallow Water and Euler Equations......Page 552
23.8 Using CLAWPACK on Quadrilateral Grids......Page 553
23.9 Boundary Conditions......Page 556
Bibliography......Page 557
Index......Page 575