Numerical solution of hyperbolic partial differential equations:
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2009
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| Beschreibung: | XXI, 597 S. graph. Darst. 1 CD-ROM (12 cm) |
| ISBN: | 9780521877275 052187727X |
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MARC
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| 100 | 1 | |a Trangenstein, John A. |d 1949- |e Verfasser |0 (DE-588)111651735 |4 aut | |
| 245 | 1 | 0 | |a Numerical solution of hyperbolic partial differential equations |c John A. Trangenstein |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2009 | |
| 300 | |a XXI, 597 S. |b graph. Darst. |e 1 CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Differential equations, Hyperbolic / Numerical solutions / Textbooks | |
| 650 | 0 | 7 | |a Hyperbolische Differentialgleichung |0 (DE-588)4131213-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Hyperbolische Differentialgleichung |0 (DE-588)4131213-2 |D s |
| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | |5 DE-604 | |
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Datensatz im Suchindex
| _version_ | 1804137152561283072 |
|---|---|
| adam_text | Contents
Preface
page
xvii
1
Introduction to Partial Differential Equations
1
2
Scalar Hyperbolic Conservation Laws
6
2.1
Linear Advection
6
2.1.1
Conservation Law on an Unbounded Domain
6
2.1.2
Integral Form of the Conservation Law
8
2.1.3
Advection-Diffusion Equation
9
2.1.4
Advection Equation on a Half-Line
10
2.1.5
Advection Equation on a Finite Interval
11
2.2
Linear Finite Difference Methods
12
2.2.1
Basics of Discretization
12
2.2.2
Explicit Upwind Differences
14
2.2.3
Programs for Explicit Upwind Differences
16
2.2.3.1
First Upwind Difference Program
16
2.2.3.2
Second Upwind Difference Program
17
2.2.3.3
Third Upwind Difference Program
18
2.2.3.4
Fourth Upwind Difference Program
20
2.2.3.5
Fifth Upwind Difference Program
21
2.2.4
Explicit Downwind Differences
23
2.2.5
Implicit Downwind Differences
24
2.2.6
Implicit Upwind Differences
25
2.2.7
Explicit Centered Differences
26
2.3
Modified Equation Analysis
30
2.3.1
Modified Equation Analysis for Explicit Upwind
Differences
30
vu
viii Contents
2.3.2
Modified Equation Analysis for Explicit Downwind
Differences
31
2.3.3
Modified Equation Analysis for Explicit Centered
Differences
32
2.3.4
Modified Equation Analysis Literature
33
2.4
Consistency, Stability and Convergence
35
2.5
Fourier Analysis of Finite Difference Schemes
38
2.5.1
Constant Coefficient Equations and Waves
39
2.5.2
Dimensionless Groups
40
2.5.3
Linear Finite Differences and Advection
41
2.5.4
Fourier Analysis of Individual Schemes
44
2.6
L2
Stability for Linear Schemes
53
2.7
Lax Equivalence Theorem
55
2.8
Measuring Accuracy and Efficiency
69
3
Nonlinear Scalar Laws
81
3.1
Nonlinear Hyperbolic Conservation Laws
81
3.1.1
Nonlinear Equations on Unbounded Domains
81
3.
3.
3.
3.
3.
3.
3.
3.
.2
Characteristics
82
.3
Development of Singularities
84
.4
Propagation of Discontinuities
85
.5
Traveling Wave Profiles
89
.6
Entropy Functions
92
.7
Oleinik Chord Condition
95
.8
Riemann Problems
97
.9
Galilean Coordinate Transformations
99
3.2
Casestudies
102
3.2.1
Traffic Flow
102
3.2.2
Miscible
Displacement Model
103
3.2.3
Buckley-Leverett Model
105
3.3
First-Order Finite Difference Methods 111
3.3.1
Explicit Upwind Differences
111
3.3.2
Lax-Friedrichs Scheme
112
3.3.3
Timestep Selection
117
3.3.4
Rusanov s Scheme
118
3.3.5
Godunov s Scheme
120
3.3.6
Comparison of Lax-Friedrichs, Godunov and Rusanov
124
3.4
Nonreflecting Boundary Conditions
125
3.5
Lax-Wendroff Process
129
3.6
Other Second Order Schemes
132
Contents ix
Nonlinear Hyperbolic Systems
135
4.1
Theory of Hyperbolic Systems
135
4.1.1
Hyperbolicky
and Characteristics
135
4.1.2
Linear Systems
139
4.1.3
Frames of Reference
140
4.1.3.1
Useful Identities
141
4.1.3.2
Change of Frame of Reference for Conservation
Laws
143
4.1.3.3
Change of Frame of Reference for Propagating
Discontinuities
145
4.1.4
Rankine-Hugoniot Jump Condition
146
4.1.5
Lax Admissibility Conditions
150
4.1.6
Asymptotic Behavior of Hugoniot Loci
152
4.1.7
Centered Rarefactions
156
4.1.8
Riemann Problems
159
4.1.9
Riemann Problem for Linear Systems
159
4.1.10
Riemann Problem for Shallow Water
162
4.1.11
Entropy Functions
164
4.2
Upwind Schemes
176
4.2.1
Lax-Friedrichs Scheme
176
4.2.2
Rusanov Scheme
179
4.2.3
Godunov Scheme
179
4.3
Case Study: Maxwell s Equations
183
4.3.1
Conservation Laws
183
4.3.2
Characteristic Analysis
184
4.4
Case Study: Gas Dynamics
186
4.4.1
Conservation Laws
187
4.4.2
Thermodynamics
187
4.4.3
Characteristic Analysis
188
4.4.4
Entropy Function
190
4.4.5
Centered Rarefaction Curves
192
4.4.6
Jump Conditions
194
4.4.7
Riemann Problem
200
4.4.8
Reflecting Walls
205
4.5
Case Study: Magnetohydrodynamics
(MHD)
208
4.5.1
Conservation Laws
208
4.5.2
Characteristic Analysis
209
4.5.3
Entropy Function
218
4.5.4
Centered Rarefaction Curves
218
4.5.5
Jump Conditions
220
Contents
4.6
Case Study:
Finite
Deformation in
Elastic
Solids
221
4.6.1 Eulerian
Formulation of Equations of Motion for Solids
221
4.6.2
Lagrangian Formulation of Equations of Motion for Solids
222
4.6.3
Constitutive Laws
223
4.6.4
Conservation Form of the Equations of Motion for Solids
225
4.6.5
Jump Conditions for Isothermal Solids
226
4.6.6
Characteristic Analysis for Solids
227
4.7
Case Study: Linear Elasticity
233
4.8
Case Study: Vibrating String
235
4.8.1
Conservation Laws
235
4.8.2
Characteristic Analysis
237
4.8.3
Jump Conditions
238
4.8.4
Lax Admissibility Conditions
240
4.8.5
Entropy Function
240
4.8.6
Wave Families for Concave Tension
241
4.8.7
Wave Family Intersections
245
4.8.8
Riemann Problem Solution
249
4.9
Case Study: Plasticity
255
4.9.1
Lagrangian Equations of Motion
255
4.9.2
Constitutive Laws
256
4.9.3
Centered Rarefactions
258
4.9.4
Hugoniot Loci
259
4.9.5
Entropy Function
261
4.9.6
Riemann Problem
261
4.10
Case Study: Polymer Model
267
4.10.1
Constitutive Laws
268
4.10.2
Characteristic Analysis
269
4.10.3
Jump Conditions
270
4.10.4
Riemann Problem Solution
271
4.11
Case Study: Three-Phase Buckley-Leverett Flow
273
4.11.1
Constitutive Models
273
4.11.2
Characteristic Analysis
275
4.11.3
Umbilic Point
276
4.11.4
Elliptic Regions
276
4.12
Case Study: Schaeffer-Schechter-Shearer System
277
4.13
Approximate Riemann Solvers
283
4.13.1
Design of Approximate Riemann Solvers
283
4.13.2
Artificial Diffusion
290
4.13.3
Rusanov Solver
292
4.13.4
Weak Wave Riemann Solver
293
Contents xi
4.13.5 Colella-Glaz Riemann
Solver
295
4.13.6 Osher-Solomon Riemann
Solver
297
4.13.7 Bell-Colella-Trangenstein
Approximate
Riemann
Problem
Solver
298
4.13.8
Roe
Riemann
Solver
303
4.13.9 Harten-Hyman
Modification of the Roe Solver
312
4.13.10
Harten-Lax-van Leer Scheme
314
4.13.11
HLL Solvers with Two Intermediate States
316
4.13.12
Approximate Riemann Solver Recommendations
319
Methods for Scalar Laws
326
5.1
Convergence
326
5.1.1
Consistency and Order
326
5.1.2
Linear Methods and Stability
328
5.1.3
Convergence of Linear Methods
330
5.2
Entropy Conditions and Difference Approximations
331
5.2.1
Bounded Convergence
331
5.2.2
Monotone Schemes
341
5.3
Nonlinear Stability
353
5.3.1
Total Variation
353
5.3.2
Total Variation Stability
354
5.3.3
Other Stability Notions
357
5.4
Propagation of Numerical Discontinuities
359
5.5 Monotonie
Schemes
361
5.5.1
Smoothness Monitor
361
5.5.2
Monotonizations
362
5.5.3
MUSCL Scheme
364
5.6
Discrete Entropy Conditions
367
5.7
E-Schemes
368
5.8
Total Variation Diminishing Schemes
370
5.8.1
Sufficient Conditions for Diminishing Total Variation
370
5.8.2
Higher-Order TVD Schemes for Linear Advection
374
5.8.3
Extension to Nonlinear Scalar Conservation Laws
378
5.9
Slope-Limiter Schemes
382
5.9.1
Exact Integration for Constant Velocity
383
5.9.2
Piecewise Linear Reconstruction
385
5.9.3
Temporal Quadrature for Flux Integrals
387
5.9.4
Characteristic Tracing
388
5.9.5
Flux Evaluation
389
5.9.6
Non-Reflecting Boundaries with the MUSCL Scheme
390
xii Contents
5.10
Wave Propagation Slope
Limiter
Schemes
391
5.10.1
Cell-Centered Wave Propagation
391
5.10.2
Side-Centered Wave Propagation
394
5.11
Higher-Order Extensions of the Lax-Friedrichs
Scheme
395
5.12
Piecewise Parabolic Method
402
5.13
Essentially Non-Oscillatory Schemes
408
5.14
Discontinuous Galerkin Methods
412
5.14.1
Weak Formulation
412
5.14.2
Basis Functions
413
5.14.3
Numerical Quadrature
414
5.14.4
Initial Data
415
5.14.5 Limiters 416
5.14.6 Timestep
Selection
417
5.15
Casestudies
418
5.15.1
Case Study: Linear Advection
418
5.15.2
Case Study: Burgers Equation
422
5.15.3
Case Study: Traffic Flow
426
5.15.4
Case Study: Buckley-Leverett Model
427
6
Methods for Hyperbolic Systems
432
6.1
First-Order Schemes for Nonlinear Systems
432
6.1.1
Lax-Friedrichs Method
432
6.1.2
Random Choice Method
433
6.1.3
Godunov s Method
433
6.1.3.1
Godunov s Method with the Rusanov Flux
434
6.1.3.2
Godunov s Method with the
Harten-Lax-vanLeer (HLL) Solver
435
6.1.3.3
Godunov s Method with the Harten-Hyman
Fix for Roe s Solver
436
6.2
Second-Order Schemes for Nonlinear Systems
438
6.2.1
Lax-Wendroff Method
438
6.2.2
MacCormack s Method
439
6.2.3
Higher-Order Lax-Friedrichs Schemes
439
6.2.4
TVD Methods
443
6.2.5
MUSCL
447
6.2.6
Wave Propagation Methods
448
6.2.7
PPM
450
6.2.8
ENO
452
6.2.9
Discontinuous Galerkin Method
453
Case
Studies
6.3.1
Wave Equation
6.3.2
Shallow Water
6.3.3
Gas Dynamics
6.3.4
MHD
6.3.5
Nonlinear Elasticity
6.3.6
Cristescu s Vibrating String
6.3.7
Plasticity
6.3.8
Polymer Model
6.3.9
Schaeffer-Schechter-Shearer Model
Contents
xiii
6.3
Case Studies
456
456
456
459
461
461
461
464
467
470
7
Methods in Multiple Dimensions
474
7.1
Numerical Methods in Two Dimensions
474
7.1.1
Operator Splitting
474
7.1.2
Donor Cell Methods
476
7.1.2.1
Traditional Donor Cell Upwind Method
478
7.1.2.2
First-Order Corner Transport Upwind Method
479
7.1.2.3
Wave Propagation Form of First-Order Corner
Transport Upwind
483
7.1.2.4
Second-Order Corner Transport Upwind
Method
485
7.1.3
Wave Propagation
488
7.1.4
2D Lax-Friedrichs
489
7.1.4.1
First-Order Lax-Friedrichs
490
7.1.4.2
Second-Order Lax-Friedrichs
491
7.1.5
Multidimensional
ENO
494
7.1.6
Discontinuous Galerkin Method on Rectangles
494
7.2
Riemann Problems in Two Dimensions
498
7.2.1
Burgers Equation
498
7.2.2
Shallow Water
500
7.2.3
Gas Dynamics
503
7.3
Numerical Methods in Three Dimensions
506
7.3.1
Operator Splitting
506
7.3.2
Donor Cell Methods
508
7.3.3
Corner Transport Upwind Scheme
510
7.3.3.1
Linear Advection with Positive Velocity
513
7.3.3.2
Linear Advection with Arbitrary Velocity
517
7.3.3.3
General Nonlinear Problems
518
7.3.3.4
Second-Order Corner Transport Upwind
519
7.3.4
Wave Propagation
521
xiv Contents
7.4
Curvilinear Coordinates
521
7.4.1
Coordinate Transformations
522
7.4.2
Spherical Coordinates
523
7.4.2.1
Case Study: Eulerian Gas Dynamics in Spherical
Coordinates
527
7.4.2.2
Case Study: Lagrangian Solid Mechanics in
Spherical Coordinates
529
7.4.3
Cylindrical Coordinates
533
7.4.3.1
Case Study: Eulerian Gas Dynamics in
Cylindrical Coordinates
537
7.4.3.2
Case Study: Lagrangian Solid Mechanics in
Cylindrical Coordinates
539
7.5
Source Terms
542
7.6
Geometric Flexibility
542
Adaptive Mesh Refinement
544
8.1
Localized Phenomena
544
8.2
Basic Assumptions
546
8.3
Outline of the Algorithm
547
8.3.1
Timestep Selection
548
8.3.2
Advancing the Patches
549
8.3.2.1
Boundary Data
549
8.3.2.2
Flux Computation
550
8.3.2.3
Time Integration
552
8.3.3
Regridding
553
8.3.3.1
Proper Nesting
553
8.3.3.2
Tagging Cells for Refinement
556
8.3.3.3
Tag Buffering
559
8.3.3.4
Logically Rectangular Organization
559
8.3.3.5
Initializing Data after Regridding
559
8.3.4
Refluxing
560
8.3.5
Upscaling
560
8.3.6
Initialization
561
8.4
Object Oriented Programming
561
8.4.1
Programming Languages
562
8.4.2
AMR Classes
563
8.4.2.1
Geometric Indices
563
8.4.2.2
Boxes
567
8.4.2.3
Data Pointers
569
8.4.2.4
Lists
569
8.4.2.5
FlowVariables
8.4.2.6
Timesteps
8.4.2.7
TagBoxes
8.4.2.8
DataBoxes
8.4.2.9
EOSModels
8.4.2.10
Patch
8.4.2.11
Level
8.5
ScalarLaw
Example
8.5.1
ScalarLaw
Constructor
8.5.2
initialize
8.5.3
stableDt
8.5.4
stuftModelGhost
8.5.5
stuffBoxGhost
8.5.6
computeFluxes
8.5.7
conservativeDifference
8.5.8
findErrorCells
8.5.9
Numerical Example
8.6
Lineai
г
Elasticity Example
8.7
Gas Dynamics
Examples
Bibliography
Index
Contents
xv
570
571
571
571
572
572
573
573
576
576
577
577
578
578
579
579
579
580
581
584
593
Numerical Solution of Hyperbolic Partial Differential Equations is
а
new type of graduate
textbook, with both print and interactive electronic components (on CD). It is a
comprehensive presentation of modern shock-capturing methods, including both finite
volume and finite element methods, covering the theory of hyperbolic conservation laws
and the theory of the numerical methods.
Classical techniques for judging the qualitative performance of the schemes, such as
modified equation analysis and Fourier analysis, are used to motivate the development of
classical higher-order methods (the Lax-Wendroff process) and to prove results such as
the Lax Equivalence Theorem.
The range of applications (shallow water, compressible gas dynamics,
magnetohydrodynamics, finite deformation in solids, plasticity, polymer flooding, and
water/gas injection in oil recovery) is broad enough to engage most engineering
disciplines and many areas of applied mathematics.
The solution of the Riemann problems for these applications is developed, so that the
reader can use the theory to develop test problems for the methods, especially to measure
errors for comparisons of accuracy and efficiency. The numerical methods involve a variety
of important approaches, such as MUSCL and PPM, TVD, wave propagation, Lax-Friedrichs
(aka central schemes),
ENO,
and discontinuous Galerkin; all of these are discussed in one
and multiple spatial dimensions. Since many of these methods depend on Riemann
solvers, there is extensive discussion of the basic design principles of approximate
Riemann solvers, and several computationally useful techniques. The final chapter
contains a discussion of adaptive mesh refinement via structured grids.
The accompanying CD contains a hyperlinked version of the text, which provides access to
computer codes
forali
of the text figures. Through this electronic text students can:
•
See the codes and run them, choosing their own input parameters interactively
•
View the online numerical results as movies
•
Gain an appreciation for both the dynamics of the problem application and the growth
of numerical errors
•
Download and modify the code for use with other applications
•
Study the code to learn how to structure their programs for modularity and ease of
debugging
John A. Trangenstein is Professor of Mathematics at Duke University, North Carolina.
|
| adam_txt |
Contents
Preface
page
xvii
1
Introduction to Partial Differential Equations
1
2
Scalar Hyperbolic Conservation Laws
6
2.1
Linear Advection
6
2.1.1
Conservation Law on an Unbounded Domain
6
2.1.2
Integral Form of the Conservation Law
8
2.1.3
Advection-Diffusion Equation
9
2.1.4
Advection Equation on a Half-Line
10
2.1.5
Advection Equation on a Finite Interval
11
2.2
Linear Finite Difference Methods
12
2.2.1
Basics of Discretization
12
2.2.2
Explicit Upwind Differences
14
2.2.3
Programs for Explicit Upwind Differences
16
2.2.3.1
First Upwind Difference Program
16
2.2.3.2
Second Upwind Difference Program
17
2.2.3.3
Third Upwind Difference Program
18
2.2.3.4
Fourth Upwind Difference Program
20
2.2.3.5
Fifth Upwind Difference Program
21
2.2.4
Explicit Downwind Differences
23
2.2.5
Implicit Downwind Differences
24
2.2.6
Implicit Upwind Differences
25
2.2.7
Explicit Centered Differences
26
2.3
Modified Equation Analysis
30
2.3.1
Modified Equation Analysis for Explicit Upwind
Differences
30
vu
viii Contents
2.3.2
Modified Equation Analysis for Explicit Downwind
Differences
31
2.3.3
Modified Equation Analysis for Explicit Centered
Differences
32
2.3.4
Modified Equation Analysis Literature
33
2.4
Consistency, Stability and Convergence
35
2.5
Fourier Analysis of Finite Difference Schemes
38
2.5.1
Constant Coefficient Equations and Waves
39
2.5.2
Dimensionless Groups
40
2.5.3
Linear Finite Differences and Advection
41
2.5.4
Fourier Analysis of Individual Schemes
44
2.6
L2
Stability for Linear Schemes
53
2.7
Lax Equivalence Theorem
55
2.8
Measuring Accuracy and Efficiency
69
3
Nonlinear Scalar Laws
81
3.1
Nonlinear Hyperbolic Conservation Laws
81
3.1.1
Nonlinear Equations on Unbounded Domains
81
3.
3.
3.
3.
3.
3.
3.
3.
.2
Characteristics
82
.3
Development of Singularities
84
.4
Propagation of Discontinuities
85
.5
Traveling Wave Profiles
89
.6
Entropy Functions
92
.7
Oleinik Chord Condition
95
.8
Riemann Problems
97
.9
Galilean Coordinate Transformations
99
3.2
Casestudies
102
3.2.1
Traffic Flow
102
3.2.2
Miscible
Displacement Model
103
3.2.3
Buckley-Leverett Model
105
3.3
First-Order Finite Difference Methods 111
3.3.1
Explicit Upwind Differences
111
3.3.2
Lax-Friedrichs Scheme
112
3.3.3
Timestep Selection
117
3.3.4
Rusanov's Scheme
118
3.3.5
Godunov's Scheme
120
3.3.6
Comparison of Lax-Friedrichs, Godunov and Rusanov
124
3.4
Nonreflecting Boundary Conditions
125
3.5
Lax-Wendroff Process
129
3.6
Other Second Order Schemes
132
Contents ix
Nonlinear Hyperbolic Systems
135
4.1
Theory of Hyperbolic Systems
135
4.1.1
Hyperbolicky
and Characteristics
135
4.1.2
Linear Systems
139
4.1.3
Frames of Reference
140
4.1.3.1
Useful Identities
141
4.1.3.2
Change of Frame of Reference for Conservation
Laws
143
4.1.3.3
Change of Frame of Reference for Propagating
Discontinuities
145
4.1.4
Rankine-Hugoniot Jump Condition
146
4.1.5
Lax Admissibility Conditions
150
4.1.6
Asymptotic Behavior of Hugoniot Loci
152
4.1.7
Centered Rarefactions
156
4.1.8
Riemann Problems
159
4.1.9
Riemann Problem for Linear Systems
159
4.1.10
Riemann Problem for Shallow Water
162
4.1.11
Entropy Functions
164
4.2
Upwind Schemes
176
4.2.1
Lax-Friedrichs Scheme
176
4.2.2
Rusanov Scheme
179
4.2.3
Godunov Scheme
179
4.3
Case Study: Maxwell's Equations
183
4.3.1
Conservation Laws
183
4.3.2
Characteristic Analysis
184
4.4
Case Study: Gas Dynamics
186
4.4.1
Conservation Laws
187
4.4.2
Thermodynamics
187
4.4.3
Characteristic Analysis
188
4.4.4
Entropy Function
190
4.4.5
Centered Rarefaction Curves
192
4.4.6
Jump Conditions
194
4.4.7
Riemann Problem
200
4.4.8
Reflecting Walls
205
4.5
Case Study: Magnetohydrodynamics
(MHD)
208
4.5.1
Conservation Laws
208
4.5.2
Characteristic Analysis
209
4.5.3
Entropy Function
218
4.5.4
Centered Rarefaction Curves
218
4.5.5
Jump Conditions
220
Contents
4.6
Case Study:
Finite
Deformation in
Elastic
Solids
221
4.6.1 Eulerian
Formulation of Equations of Motion for Solids
221
4.6.2
Lagrangian Formulation of Equations of Motion for Solids
222
4.6.3
Constitutive Laws
223
4.6.4
Conservation Form of the Equations of Motion for Solids
225
4.6.5
Jump Conditions for Isothermal Solids
226
4.6.6
Characteristic Analysis for Solids
227
4.7
Case Study: Linear Elasticity
233
4.8
Case Study: Vibrating String
235
4.8.1
Conservation Laws
235
4.8.2
Characteristic Analysis
237
4.8.3
Jump Conditions
238
4.8.4
Lax Admissibility Conditions
240
4.8.5
Entropy Function
240
4.8.6
Wave Families for Concave Tension
241
4.8.7
Wave Family Intersections
245
4.8.8
Riemann Problem Solution
249
4.9
Case Study: Plasticity
255
4.9.1
Lagrangian Equations of Motion
255
4.9.2
Constitutive Laws
256
4.9.3
Centered Rarefactions
258
4.9.4
Hugoniot Loci
259
4.9.5
Entropy Function
261
4.9.6
Riemann Problem
261
4.10
Case Study: Polymer Model
267
4.10.1
Constitutive Laws
268
4.10.2
Characteristic Analysis
269
4.10.3
Jump Conditions
270
4.10.4
Riemann Problem Solution
271
4.11
Case Study: Three-Phase Buckley-Leverett Flow
273
4.11.1
Constitutive Models
273
4.11.2
Characteristic Analysis
275
4.11.3
Umbilic Point
276
4.11.4
Elliptic Regions
276
4.12
Case Study: Schaeffer-Schechter-Shearer System
277
4.13
Approximate Riemann Solvers
283
4.13.1
Design of Approximate Riemann Solvers
283
4.13.2
Artificial Diffusion
290
4.13.3
Rusanov Solver
292
4.13.4
Weak Wave Riemann Solver
293
Contents xi
4.13.5 Colella-Glaz Riemann
Solver
295
4.13.6 Osher-Solomon Riemann
Solver
297
4.13.7 Bell-Colella-Trangenstein
Approximate
Riemann
Problem
Solver
298
4.13.8
Roe
Riemann
Solver
303
4.13.9 Harten-Hyman
Modification of the Roe Solver
312
4.13.10
Harten-Lax-van Leer Scheme
314
4.13.11
HLL Solvers with Two Intermediate States
316
4.13.12
Approximate Riemann Solver Recommendations
319
Methods for Scalar Laws
326
5.1
Convergence
326
5.1.1
Consistency and Order
326
5.1.2
Linear Methods and Stability
328
5.1.3
Convergence of Linear Methods
330
5.2
Entropy Conditions and Difference Approximations
331
5.2.1
Bounded Convergence
331
5.2.2
Monotone Schemes
341
5.3
Nonlinear Stability
353
5.3.1
Total Variation
353
5.3.2
Total Variation Stability
354
5.3.3
Other Stability Notions
357
5.4
Propagation of Numerical Discontinuities
359
5.5 Monotonie
Schemes
361
5.5.1
Smoothness Monitor
361
5.5.2
Monotonizations
362
5.5.3
MUSCL Scheme
364
5.6
Discrete Entropy Conditions
367
5.7
E-Schemes
368
5.8
Total Variation Diminishing Schemes
370
5.8.1
Sufficient Conditions for Diminishing Total Variation
370
5.8.2
Higher-Order TVD Schemes for Linear Advection
374
5.8.3
Extension to Nonlinear Scalar Conservation Laws
378
5.9
Slope-Limiter Schemes
382
5.9.1
Exact Integration for Constant Velocity
383
5.9.2
Piecewise Linear Reconstruction
385
5.9.3
Temporal Quadrature for Flux Integrals
387
5.9.4
Characteristic Tracing
388
5.9.5
Flux Evaluation
389
5.9.6
Non-Reflecting Boundaries with the MUSCL Scheme
390
xii Contents
5.10
Wave Propagation Slope
Limiter
Schemes
391
5.10.1
Cell-Centered Wave Propagation
391
5.10.2
Side-Centered Wave Propagation
394
5.11
Higher-Order Extensions of the Lax-Friedrichs
Scheme
395
5.12
Piecewise Parabolic Method
402
5.13
Essentially Non-Oscillatory Schemes
408
5.14
Discontinuous Galerkin Methods
412
5.14.1
Weak Formulation
412
5.14.2
Basis Functions
413
5.14.3
Numerical Quadrature
414
5.14.4
Initial Data
415
5.14.5 Limiters 416
5.14.6 Timestep
Selection
417
5.15
Casestudies
418
5.15.1
Case Study: Linear Advection
418
5.15.2
Case Study: Burgers' Equation
422
5.15.3
Case Study: Traffic Flow
426
5.15.4
Case Study: Buckley-Leverett Model
427
6
Methods for Hyperbolic Systems
432
6.1
First-Order Schemes for Nonlinear Systems
432
6.1.1
Lax-Friedrichs Method
432
6.1.2
Random Choice Method
433
6.1.3
Godunov's Method
433
6.1.3.1
Godunov's Method with the Rusanov Flux
434
6.1.3.2
Godunov's Method with the
Harten-Lax-vanLeer (HLL) Solver
435
6.1.3.3
Godunov's Method with the Harten-Hyman
Fix for Roe's Solver
436
6.2
Second-Order Schemes for Nonlinear Systems
438
6.2.1
Lax-Wendroff Method
438
6.2.2
MacCormack's Method
439
6.2.3
Higher-Order Lax-Friedrichs Schemes
439
6.2.4
TVD Methods
443
6.2.5
MUSCL
447
6.2.6
Wave Propagation Methods
448
6.2.7
PPM
450
6.2.8
ENO
452
6.2.9
Discontinuous Galerkin Method
453
Case
Studies
6.3.1
Wave Equation
6.3.2
Shallow Water
6.3.3
Gas Dynamics
6.3.4
MHD
6.3.5
Nonlinear Elasticity
6.3.6
Cristescu's Vibrating String
6.3.7
Plasticity
6.3.8
Polymer Model
6.3.9
Schaeffer-Schechter-Shearer Model
Contents
xiii
6.3
Case Studies
456
456
456
459
461
461
461
464
467
470
7
Methods in Multiple Dimensions
474
7.1
Numerical Methods in Two Dimensions
474
7.1.1
Operator Splitting
474
7.1.2
Donor Cell Methods
476
7.1.2.1
Traditional Donor Cell Upwind Method
478
7.1.2.2
First-Order Corner Transport Upwind Method
479
7.1.2.3
Wave Propagation Form of First-Order Corner
Transport Upwind
483
7.1.2.4
Second-Order Corner Transport Upwind
Method
485
7.1.3
Wave Propagation
488
7.1.4
2D Lax-Friedrichs
489
7.1.4.1
First-Order Lax-Friedrichs
490
7.1.4.2
Second-Order Lax-Friedrichs
491
7.1.5
Multidimensional
ENO
494
7.1.6
Discontinuous Galerkin Method on Rectangles
494
7.2
Riemann Problems in Two Dimensions
498
7.2.1
Burgers' Equation
498
7.2.2
Shallow Water
500
7.2.3
Gas Dynamics
503
7.3
Numerical Methods in Three Dimensions
506
7.3.1
Operator Splitting
506
7.3.2
Donor Cell Methods
508
7.3.3
Corner Transport Upwind Scheme
510
7.3.3.1
Linear Advection with Positive Velocity
513
7.3.3.2
Linear Advection with Arbitrary Velocity
517
7.3.3.3
General Nonlinear Problems
518
7.3.3.4
Second-Order Corner Transport Upwind
519
7.3.4
Wave Propagation
521
xiv Contents
7.4
Curvilinear Coordinates
521
7.4.1
Coordinate Transformations
522
7.4.2
Spherical Coordinates
523
7.4.2.1
Case Study: Eulerian Gas Dynamics in Spherical
Coordinates
527
7.4.2.2
Case Study: Lagrangian Solid Mechanics in
Spherical Coordinates
529
7.4.3
Cylindrical Coordinates
533
7.4.3.1
Case Study: Eulerian Gas Dynamics in
Cylindrical Coordinates
537
7.4.3.2
Case Study: Lagrangian Solid Mechanics in
Cylindrical Coordinates
539
7.5
Source Terms
542
7.6
Geometric Flexibility
542
Adaptive Mesh Refinement
544
8.1
Localized Phenomena
544
8.2
Basic Assumptions
546
8.3
Outline of the Algorithm
547
8.3.1
Timestep Selection
548
8.3.2
Advancing the Patches
549
8.3.2.1
Boundary Data
549
8.3.2.2
Flux Computation
550
8.3.2.3
Time Integration
552
8.3.3
Regridding
553
8.3.3.1
Proper Nesting
553
8.3.3.2
Tagging Cells for Refinement
556
8.3.3.3
Tag Buffering
559
8.3.3.4
Logically Rectangular Organization
559
8.3.3.5
Initializing Data after Regridding
559
8.3.4
Refluxing
560
8.3.5
Upscaling
560
8.3.6
Initialization
561
8.4
Object Oriented Programming
561
8.4.1
Programming Languages
562
8.4.2
AMR Classes
563
8.4.2.1
Geometric Indices
563
8.4.2.2
Boxes
567
8.4.2.3
Data Pointers
569
8.4.2.4
Lists
569
8.4.2.5
FlowVariables
8.4.2.6
Timesteps
8.4.2.7
TagBoxes
8.4.2.8
DataBoxes
8.4.2.9
EOSModels
8.4.2.10
Patch
8.4.2.11
Level
8.5
ScalarLaw
Example
8.5.1
ScalarLaw
Constructor
8.5.2
initialize
8.5.3
stableDt
8.5.4
stuftModelGhost
8.5.5
stuffBoxGhost
8.5.6
computeFluxes
8.5.7
conservativeDifference
8.5.8
findErrorCells
8.5.9
Numerical Example
8.6
Lineai
г
Elasticity Example
8.7
Gas Dynamics
Examples
Bibliography
Index
Contents
xv
570
571
571
571
572
572
573
573
576
576
577
577
578
578
579
579
579
580
581
584
593
Numerical Solution of Hyperbolic Partial Differential Equations is
а
new type of graduate
textbook, with both print and interactive electronic components (on CD). It is a
comprehensive presentation of modern shock-capturing methods, including both finite
volume and finite element methods, covering the theory of hyperbolic conservation laws
and the theory of the numerical methods.
Classical techniques for judging the qualitative performance of the schemes, such as
modified equation analysis and Fourier analysis, are used to motivate the development of
classical higher-order methods (the Lax-Wendroff process) and to prove results such as
the Lax Equivalence Theorem.
The range of applications (shallow water, compressible gas dynamics,
magnetohydrodynamics, finite deformation in solids, plasticity, polymer flooding, and
water/gas injection in oil recovery) is broad enough to engage most engineering
disciplines and many areas of applied mathematics.
The solution of the Riemann problems for these applications is developed, so that the
reader can use the theory to develop test problems for the methods, especially to measure
errors for comparisons of accuracy and efficiency. The numerical methods involve a variety
of important approaches, such as MUSCL and PPM, TVD, wave propagation, Lax-Friedrichs
(aka central schemes),
ENO,
and discontinuous Galerkin; all of these are discussed in one
and multiple spatial dimensions. Since many of these methods depend on Riemann
solvers, there is extensive discussion of the basic design principles of approximate
Riemann solvers, and several computationally useful techniques. The final chapter
contains a discussion of adaptive mesh refinement via structured grids.
The accompanying CD contains a hyperlinked version of the text, which provides access to
computer codes
forali
of the text figures. Through this electronic text students can:
•
See the codes and run them, choosing their own input parameters interactively
•
View the online numerical results as movies
•
Gain an appreciation for both the dynamics of the problem application and the growth
of numerical errors
•
Download and modify the code for use with other applications
•
Study the code to learn how to structure their programs for modularity and ease of
debugging
John A. Trangenstein is Professor of Mathematics at Duke University, North Carolina. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Trangenstein, John A. 1949- |
| author_GND | (DE-588)111651735 |
| author_facet | Trangenstein, John A. 1949- |
| author_role | aut |
| author_sort | Trangenstein, John A. 1949- |
| author_variant | j a t ja jat |
| building | Verbundindex |
| bvnumber | BV022886408 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377 |
| callnumber-search | QA377 |
| callnumber-sort | QA 3377 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 560 SK 920 |
| ctrlnum | (OCoLC)145388396 (DE-599)BSZ268038988 |
| dewey-full | 518.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.64 |
| dewey-search | 518.64 |
| dewey-sort | 3518.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| edition | 1. publ. |
| format | Book |
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| illustrated | Illustrated |
| index_date | 2024-07-02T18:52:05Z |
| indexdate | 2024-07-09T21:07:45Z |
| institution | BVB |
| isbn | 9780521877275 052187727X |
| language | English |
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| physical | XXI, 597 S. graph. Darst. 1 CD-ROM (12 cm) |
| publishDate | 2009 |
| publishDateSearch | 2009 |
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| publisher | Cambridge Univ. Press |
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| spelling | Trangenstein, John A. 1949- Verfasser (DE-588)111651735 aut Numerical solution of hyperbolic partial differential equations John A. Trangenstein 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2009 XXI, 597 S. graph. Darst. 1 CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Differential equations, Hyperbolic / Numerical solutions / Textbooks Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Hyperbolische Differentialgleichung (DE-588)4131213-2 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091313&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091313&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Trangenstein, John A. 1949- Numerical solution of hyperbolic partial differential equations Differential equations, Hyperbolic / Numerical solutions / Textbooks Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4131213-2 (DE-588)4128130-5 |
| title | Numerical solution of hyperbolic partial differential equations |
| title_auth | Numerical solution of hyperbolic partial differential equations |
| title_exact_search | Numerical solution of hyperbolic partial differential equations |
| title_exact_search_txtP | Numerical solution of hyperbolic partial differential equations |
| title_full | Numerical solution of hyperbolic partial differential equations John A. Trangenstein |
| title_fullStr | Numerical solution of hyperbolic partial differential equations John A. Trangenstein |
| title_full_unstemmed | Numerical solution of hyperbolic partial differential equations John A. Trangenstein |
| title_short | Numerical solution of hyperbolic partial differential equations |
| title_sort | numerical solution of hyperbolic partial differential equations |
| topic | Differential equations, Hyperbolic / Numerical solutions / Textbooks Hyperbolische Differentialgleichung (DE-588)4131213-2 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Differential equations, Hyperbolic / Numerical solutions / Textbooks Hyperbolische Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091313&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016091313&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT trangensteinjohna numericalsolutionofhyperbolicpartialdifferentialequations |