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Tables of Contents for Finite Volume Methods for Hyperbolic Systems
Chapter/Section Title
Page #
Page Count
Preface
xvii
Introduction
1
14
Conservation Laws
3
2
Finite Volume Methods
5
1
Multidimensional Problems
6
1
Linear Waves and Discontinuous Media
7
1
Clawpack Software
8
1
References
9
1
Notation
10
5
Part I Linear Equations
Conservation Laws and Differential Equations
15
32
The Advection Equation
17
3
Diffusion and the Advection--Diffusion Equation
20
1
The Heat Equation
21
1
Capacity Functions
22
1
Source Terms
22
1
Nonlinear Equations in Fluid Dynamics
23
3
Linear Acoustics
26
3
Sound Waves
29
2
Hyperbolicity of Linear Systems
31
2
Variable-Coefficient Hyperbolic Systems
33
1
Hyperbolicity of Quasilinear and Nonlinear Systems
34
1
Solid Mechanics and Elastic Waves
35
6
Lagrangian Gas Dynamics and the p-System
41
2
Electromagnetic Waves
43
4
Exercises
46
1
Characteristics and Riemann Problems for Linear Hyperbolic Equations
47
17
Solution to the Cauchy Problem
47
1
Superposition of Waves and Characteristic Variables
48
1
Left Eigenvectors
49
1
Simple Waves
49
1
Acoustics
49
1
Domain of Dependence and Range of Influence
50
2
Discontinuous Solutions
52
1
The Riemann Problem for a Linear System
52
3
The Phase Plane for Systems of Two Equations
55
2
Coupled Acoustics and Advection
57
2
Initial--Boundary-Value Problems
59
5
Exercises
62
2
Finite Volume Methods
64
23
General Formulation for Conservation Laws
64
2
A Numerical Flux for the Diffusion Equation
66
1
Necessary Components for Convergence
67
1
The CFL Condition
68
3
An Unstable Flux
71
1
The Lax--Friedrichs Method
71
1
The Richtmyer Two-Step Lax--Wendroff Method
72
1
Upwind Methods
72
1
The Upwind Method for Advection
73
3
Godunov's Method for Linear Systems
76
2
The Numerical Flux Function for Godunov's Method
78
1
The Wave-Propagation Form of Godunov's Method
78
5
Flux-Difference vs. Flux-Vector Splitting
83
1
Roe's Method
84
3
Exercises
85
2
Introduction to the Clawpack Software
87
13
Basic Framework
87
2
Obtaining Clawpack
89
1
Getting Started
89
2
Using Clawpack -- a Guide through example1
91
7
Other User-Supplied Routines and Files
98
1
Auxiliary Arrays and setaux.f
98
1
An Acoustics Example
99
1
Exercises
99
1
High-Resolution Methods
100
29
The Lax--Wendroff Method
100
2
The Beam--Warming Method
102
1
Preview of Limiters
103
3
The REA Algorithm with Piecewise Linear Reconstruction
106
1
Choice of Slopes
107
1
Oscillations
108
1
Total Variation
109
1
TVD Methods Based on the REA Algorithm
110
1
Slope-Limiter Methods
111
1
Flux Formulation with Piecewise Linear Reconstruction
112
2
Flux Limiters
114
1
TVD Limiters
115
3
High-Resolution Methods for Systems
118
2
Implementation
120
1
Extension to Nonlinear Systems
121
1
Capacity-Form Differencing
122
1
Nonuniform Grids
123
6
Exercises
127
2
Boundary Conditions and Ghost Cells
129
10
Periodic Boundary Conditions
130
1
Advection
130
3
Acoustics
133
6
Exercises
138
1
Convergence, Accuracy, and Stability
139
19
Convergence
139
2
One-Step and Local Truncation Errors
141
2
Stability Theory
143
6
Accuracy at Extrema
149
1
Order of Accuracy Isn't Everything
150
1
Modified Equations
151
4
Accuracy Near Discontinuities
155
3
Exercises
156
2
Variable-Coefficient Linear Equations
158
30
Advection in a Pipe
159
2
Finite Volume Methods
161
1
The Color Equation
162
2
The Conservative Advection Equation
164
5
Edge Velocities
169
2
Variable-Coefficient Acoustics Equations
171
1
Constant-Impedance Media
172
1
Variable Impedance
173
4
Solving the Riemann Problem for Acoustics
177
1
Transmission and Reflection Coefficients
178
1
Godunov's Method
179
2
High-Resolution Methods
181
1
Wave Limiters
181
2
Homogenization of Rapidly Varying Coefficients
183
5
Exercises
187
1
Other Approaches to High Resolution
188
15
Centered-in-Time Fluxes
188
2
Higher-Order High-Resolution Methods
190
1
Limitations of the Lax--Wendroff (Taylor Series) Approach
191
1
Semidiscrete Methods plus Time Stepping
191
7
Staggered Grids and Central Schemes
198
5
Exercises
200
3
Part II Nonlinear Equations
Nonlinear Scalar Conservation Laws
203
24
Traffic Flow
203
3
Quasilinear Form and Characteristics
206
2
Burgers' Equation
208
1
Rarefaction Waves
209
1
Compression Waves
210
1
Vanishing Viscosity
210
1
Equal-Area Rule
211
1
Shock Speed
212
1
The Rankine--Hugoniot Conditions for Systems
213
1
Similarity Solutions and Centered Rarefactions
214
1
Weak Solutions
215
1
Manipulating Conservation Laws
216
1
Nonuniqueness, Admissibility, and Entropy Conditions
216
3
Entropy Functions
219
3
Long-Time Behavior and N-Wave Decay
222
5
Exercises
224
3
Finite Volume Methods for Nonlinear Scalar Conservation Laws
227
26
Godunov's Method
227
2
Fluctuations, Waves, and Speeds
229
1
Transonic Rarefactions and an Entropy Fix
230
2
Numerical Viscosity
232
1
The Lax--Friedrichs and Local Lax--Friedrichs Methods
232
2
The Engquist--Osher Method
234
1
E-schemes
235
1
High-Resolution TVD Methods
235
2
The Importance of Conservation Form
237
2
The Lax--Wendroff Theorem
239
4
The Entropy Condition
243
1
Nonlinear Stability
244
9
Exercises
252
1
Nonlinear Systems of Conservation Laws
253
38
The Shallow Water Equations
254
5
Dam-Break and Riemann Problems
259
1
Characteristic Structure
260
2
A Two-Shock Riemann Solution
262
1
Weak Waves and the Linearized Problem
263
1
Strategy for Solving the Riemann Problem
263
1
Shock Waves and Hugoniot Loci
264
5
Simple Waves and Rarefactions
269
10
Solving the Dam-Break Problem
279
2
The General Riemann Solver for Shallow Water Equations
281
1
Shock Collision Problems
282
1
Linear Degeneracy and Contact Discontinuities
283
8
Exercises
287
4
Gas Dynamics and the Euler Equations
291
20
Pressure
291
1
Energy
292
1
The Euler Equations
293
1
Polytropic Ideal Gas
293
2
Entropy
295
3
Isothermal Flow
298
1
The Euler Equations in Primitive Variables
298
2
The Riemann Problem for the Euler Equations
300
1
Contact Discontinuities
301
1
Riemann Invariants
302
1
Solution to the Riemann Problem
302
3
The Structure of Rarefaction Waves
305
1
Shock Tubes and Riemann Problems
306
2
Multifluid Problems
308
1
Other Equations of State and Incompressible Flow
309
2
Finite Volume Methods for Nonlinear Systems
311
39
Godunov's Method
311
2
Convergence of Godunov's Method
313
1
Approximate Riemann Solvers
314
15
High-Resolution Methods for Nonlinear Systems
329
4
An Alternative Wave-Propagation Implementation of Approximate Riemann Solvers
333
2
Second-Order Accuracy
335
3
Flux-Vector Splitting
338
2
Total Variation for Systems of Equations
340
10
Exercises
348
2
Some Nonclassical Hyperbolic Problems
350
25
Nonconvex Flux Functions
350
8
Nonstrictly Hyperbolic Problems
358
4
Loss of Hyperbolicity
362
6
Spatially Varying Flux Functions
368
3
Nonconservative Nonlinear Hyperbolic Equations
371
1
Nonconservative Transport Equations
372
3
Exercises
374
1
Source Terms and Balance Laws
375
46
Fractional-Step Methods
377
1
An Advection--Reaction Equation
378
6
General Formulation of Fractional-Step Methods for Linear Problems
384
3
Strang Splitting
387
1
Accuracy of Godunov and Strang Splittings
388
1
Choice of ODE Solver
389
1
Implicit Methods, Viscous Terms, and Higher-Order Derivatives
390
1
Steady-State Solutions
391
2
Boundary Conditions for Fractional-Step Methods
393
3
Stiff and Singular Source Terms
396
1
Linear Traffic Flow with On-Ramps or Exits
396
1
Rankine--Hugoniot Jump Conditions at a Singular Source
397
1
Nonlinear Traffic Flow with On-Ramps or Exits
398
1
Accurate Solution of Quasisteady Problems
399
2
Burgers Equation with a Stiff Source Term
401
3
Numerical Difficulties with Stiff Source Terms
404
6
Relaxation Systems
410
5
Relaxation Schemes
415
6
Exercises
416
5
Part III Multidimensional Problems
Multidimensional Hyperbolic Problems
421
15
Derivation of Conservation Laws
421
2
Advection
423
1
Compressible Flow
424
1
Acoustics
425
1
Hyperbolicity
425
3
Three-Dimensional Systems
428
1
Shallow Water Equations
429
2
Euler Equations
431
2
Symmetry and Reduction of Dimension
433
3
Exercises
434
2
Multidimensional Numerical Methods
436
11
Finite Difference Methods
436
2
Finite Volume Methods and Approaches to Discretization
438
1
Fully Discrete Flux-Differencing Methods
439
4
Semidiscrete Methods with Runge--Kutta Time Stepping
443
1
Dimensional Splitting
444
3
Exercise
446
1
Multidimensional Scalar Equations
447
22
The Donor-Cell Upwind Method for Advection
447
2
The Corner-Transport Upwind Method for Advection
449
1
Wave-Propagation Implementation of the CTU Method
450
2
von Neumann Stability Analysis
452
1
The CTU Method for Variable-Coefficient Advection
453
3
High-Resolution Correction Terms
456
1
Relation to the Lax--Wendroff Method
456
1
Divergence-Free Velocity Fields
457
3
Nonlinear Scalar Conservation Laws
460
4
Convergence
464
5
Exercises
467
2
Multidimensional Systems
469
22
Constant-Coefficient Linear Systems
469
2
The Wave-Propagation Approach to Accumulating Fluxes
471
2
Clawpack Implementation
473
1
Acoustics
474
2
Acoustics in Heterogeneous Media
476
4
Transverse Riemann Solvers for Nonlinear Systems
480
1
Shallow Water Equations
480
5
Boundary Conditions
485
6
Elastic Waves
491
23
Derivation of the Elasticity Equations
492
7
Plane-Strain Equations of Two-Dimensional Elasticity
499
3
One-Dimensional Slices
502
1
Boundary Conditions
502
2
The Plane-Stress Equations and Two-Dimensional Plates
504
5
A One-Dimensional Rod
509
1
Two-Dimensional Elasticity in Heterogeneous Media
509
5
Finite Volume Methods on Quadrilateral Grids
514
21
Cell Averages and Interface Fluxes
515
2
Logically Rectangular Grids
517
1
Godunov's Method
518
1
Fluctuation Form
519
1
Advection Equations
520
5
Acoustics
525
5
Shallow Water and Euler Equations
530
1
Using Clawpack on Quadrilateral Grids
531
3
Boundary Conditions
534
1
Bibliography
535
18
Index
553