【08新书】Effective Computational Methods for Wave Propagation: Effective Computational Methods for Wave Propagation.part1.rar

Numerical Insights
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Effective Computational Methods for Wave Propagation
Nikolaos A. Kampanis, Vassilios A. Dougalis, and John A. Ekaterinaris

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Library of Congress Cataloging-in-Publication Data
Kampanis, Nikolaos A.
Effective computational methods for wave propagation / Nikolaos A.
Kampanis, John A. Ekaterinaris, Vassilios Dougalis.
p. cm. -- (Numerical insights)
Includes bibliographical references and index.
ISBN 978-1-58488-568-9 (alk. paper)
1. Wave-motion, Theory of--Data processing. 2. Electromagnetic waves--Data
processing. 3. Numerical analysis. I. Ekaterinaris, John A. II. Dougalis, Vassilios.
III. Title. IV. Series.
QA927.K255 2008
530.12’4--dc22 2007030361
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Contents
Preface 1
I Nonlinear Dispersive Waves 5
1 Numerical Simulations of Singular Solutions of the Nonlinear
Schr¨odinger Equations, Xiao-Ping Wang 7
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Dynamic Rescaling Method . . . . . . . . . . . . . . . . . . 8
1.2.1 Radially Symmetric Case . . . . . . . . . . . . . . . . 8
1.2.2 Anisotropic Dynamic Rescaling . . . . . . . . . . . . 9
1.3 Adaptive Method Based on the Iterative Grid Redistribution
(IGR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3.1 Grid Distribution Based on the Variational Principle 13
1.3.2 An Iterative Grid Redistribution Procedure . . . . . 16
1.3.3 Adaptive Procedure for Solving Nonlinear Schr¨odinger
Equations . . . . . . . . . . . . . . . . . . . . . . . . 17
1.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 Singular Solutions with Multiple Blowup Points . . . 18
1.4.2 Numerical Simulations of Self–Focusing of Ultrafast
Laser Pulses . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Ring Blowup Solutions of the NLS . . . . . . . . . . . 23
1.4.4 Keller-Seigel Equation: Complex Singularity . . . . . 28
1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References 32
2 Numerical Solution of the Nonlinear Helmholtz Equation, G.
Fibich and S. Tsynkov 37
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.1 Background . . . . . . . . . . . . . . . . . . . . . . . 37
2.1.2 The Nonlinear Helmholtz Equation . . . . . . . . . . 38
2.1.3 Transverse Boundary Conditions . . . . . . . . . . . . 40
2.1.4 Paraxial Approximation and the Nonlinear Schr¨odinger
Equation . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1.5 Solitons and Collapse . . . . . . . . . . . . . . . . . . 41
2.2 Algorithm — Continuous Formulation . . . . . . . . . . . . 42
v
vi
2.2.1 Iteration Scheme . . . . . . . . . . . . . . . . . . . . . 43
2.2.2 Separation of Variables and Boundary Conditions . . 43
2.3 Algorithm — Finite-Difference Formulation . . . . . . . . . 47
2.3.1 Fourth Order Scheme . . . . . . . . . . . . . . . . . . 47
2.3.2 Transverse Boundary Conditions . . . . . . . . . . . . 48
2.3.3 Discrete Eigenvalue Problem . . . . . . . . . . . . . . 50
2.3.4 Separation of Variables . . . . . . . . . . . . . . . . . 52
2.3.5 Properties of Eigenvalues . . . . . . . . . . . . . . . . 53
2.3.6 Nonlocal ABCs . . . . . . . . . . . . . . . . . . . . . 54
2.4 Results of Computations . . . . . . . . . . . . . . . . . . . . 55
2.4.1 Critical Case . . . . . . . . . . . . . . . . . . . . . . . 56
2.4.2 Subcritical Case . . . . . . . . . . . . . . . . . . . . . 57
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
References 61
3 Theory and Numerical Analysis of Boussinesq Systems: A
Review, V. A. Dougalis and D. E. Mitsotakis 63
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2 Derivation and Examples of Boussinesq Systems . . . . . . . 64
3.3 Well-Posedness Theory . . . . . . . . . . . . . . . . . . . . . 72
3.4 Solitary Waves . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . 84
3.6 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . 89
3.7 Boussinesq Systems in Two Space Dimensions . . . . . . . . 99
References 105
II The Helmholtz Equation and Its Paraxial Approximations
in Underwater Acoustics 111
4 Finite Element Discretization of the Helmholtz Equation in
an Underwater Acoustic Waveguide, D. A. Mitsoudis, N. A.
Kampanis and V. A. Dougalis 113
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.2 Formulation of the Problem . . . . . . . . . . . . . . . . . . 116
4.2.1 Reformulation of the Problem in a Bounded Domain 118
4.2.2 Construction of the Nonlocal Conditions at the Artificial
Boundaries . . . . . . . . . . . . . . . . . . . . . 119
4.3 The Finite Element Method . . . . . . . . . . . . . . . . . . 122
4.3.1 The Finite Element Discretization . . . . . . . . . . . 123
4.3.2 Implementation Issues . . . . . . . . . . . . . . . . . . 124
4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . 126
References 131
vii
5 Parabolic Equation Techniques in Underwater Acoustics, D.
J. Thomson and G. H. Brooke 135
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 Parabolic Approximations . . . . . . . . . . . . . . . . . . . 140
5.2.1 Standard PE . . . . . . . . . . . . . . . . . . . . . . . 140
5.2.2 Exact PE . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2.3 Propagator Approximations . . . . . . . . . . . . . . 142
5.2.4 Finite-Difference Scheme . . . . . . . . . . . . . . . . 144
5.2.5 Profile Interpolation . . . . . . . . . . . . . . . . . . . 145
5.2.6 Energy Conservation . . . . . . . . . . . . . . . . . . 146
5.2.7 Equivalent Fluid . . . . . . . . . . . . . . . . . . . . . 147
5.2.8 Initial Field . . . . . . . . . . . . . . . . . . . . . . . 148
5.2.9 Perfectly Matched Absorber . . . . . . . . . . . . . . 148
5.2.10 Propagation Example . . . . . . . . . . . . . . . . . . 149
5.3 PE-Based Matched Field Processing . . . . . . . . . . . . . 150
5.3.1 Standard Processor . . . . . . . . . . . . . . . . . . . 150
5.3.2 Backpropagated Processor . . . . . . . . . . . . . . . 152
5.3.3 Localization Example . . . . . . . . . . . . . . . . . . 153
5.4 Modal Decomposition . . . . . . . . . . . . . . . . . . . . . 155
5.4.1 Relation between PE Modes and Normal Modes . . . 157
5.4.2 Modal Excitations . . . . . . . . . . . . . . . . . . . . 158
5.4.3 Modal Phases . . . . . . . . . . . . . . . . . . . . . . 160
5.4.4 Modal Decomposition Example . . . . . . . . . . . . 160
5.4.5 Modal Beamforming . . . . . . . . . . . . . . . . . . . 162
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
References 165
6 Numerical Solution of the Parabolic Equation in Range–
Dependent Waveguides, V. A. Dougalis, N. A. Kampanis,
F. Sturm and G. E. Zouraris 175
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
6.2 Initial–Boundary Value Problems in Axially Symmetric
Range-Dependent Environments . . . . . . . . . . . . . . . . 178
6.3 Finite Element Solution of the 2D PE in a General Stratified
Waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
6.3.1 Horizontal Interface . . . . . . . . . . . . . . . . . . . 187
6.3.2 Sloping Interface . . . . . . . . . . . . . . . . . . . . . 193
6.4 Finite Element Solution of the 3D Standard PE in a General
Stratified Waveguide . . . . . . . . . . . . . . . . . . . . . . 198
6.4.1 The Initial–Boundary Value Problem for the 3D Standard
PE . . . . . . . . . . . . . . . . . . . . . . . . . 198
6.4.2 The Transformed Initial–Boundary Value Problem . . 200
6.4.3 The Numerical Scheme . . . . . . . . . . . . . . . . . 202
6.4.4 Numerical Examples . . . . . . . . . . . . . . . . . . . 203
viii
References 205
7 Exact Boundary Conditions for Acoustic PE Modeling Over
an N2-Linear Half-Space, T. W. Dawson, G. H. Brooke and
D. J. Thomson 209
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
7.2.1 PE Theory . . . . . . . . . . . . . . . . . . . . . . . . 212
7.2.2 Solution in the Lower Half-Space . . . . . . . . . . . . 214
7.2.3 Solution Details . . . . . . . . . . . . . . . . . . . . . 216
7.2.4 Complex Half-Space Profile–Attenuation . . . . . . . 217
7.3 Non-Local Boundary Conditions . . . . . . . . . . . . . . . . 217
7.3.1 General Considerations . . . . . . . . . . . . . . . . . 217
7.3.2 Narrow Angle (Tappert) PE . . . . . . . . . . . . . . 219
7.3.3 Simple Wide-Angle (Claerbout) PE . . . . . . . . . . 220
7.4 Numerical Implementation . . . . . . . . . . . . . . . . . . . 223
7.5 First-Order Claerbout Examples . . . . . . . . . . . . . . . 225
7.5.1 Modified AESD [13] Case . . . . . . . . . . . . . . . . 225
7.5.2 Modified Norda Test Cases . . . . . . . . . . . . . . . 227
7.6 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . 229
References 237
III Numerical Methods for Elastic Wave Propagation
239
8 Introduction and Orientation, P. Joly 241
9 TheMathematical Model for Elastic Wave Propagation, P.
Joly 247
9.1 Preliminary Notation . . . . . . . . . . . . . . . . . . . . . . 247
9.2 The Equations of Linear Elastodynamics . . . . . . . . . . . 249
9.2.1 The Unknowns of the Problem . . . . . . . . . . . . . 250
9.2.2 Useful Differential Operators and Green’s Formulas . 250
9.2.3 The Equations of the Problem . . . . . . . . . . . . . 251
9.3 Variational Formulation and Weak Solutions . . . . . . . . . 253
9.4 Plane Wave Propagation in Homogeneous Media . . . . . . 257
9.5 Finite Propagation Velocity . . . . . . . . . . . . . . . . . . 263
10 Finite Element Methods with Continuous Displacement, P.
Joly 267
10.1 Galerkin Approximation of Abstract Second Order Variational
Evolution Problems . . . . . . . . . . . . . . . . . . . . . . . 267
10.2 Space Approximation of Elastodynamics Equations with Lagrange
Finite Elements . . . . . . . . . . . . . . . . . . . . . 274
ix
10.3 On the Use of Quadrature Formulas . . . . . . . . . . . . . 287
10.4 The Mass Lumping Technique . . . . . . . . . . . . . . . . . 305
10.5 Time Discretization by Finite Differences . . . . . . . . . . . 313
10.6 Computational Issues . . . . . . . . . . . . . . . . . . . . . . 321
10.6.1 General Considerations . . . . . . . . . . . . . . . . . 321
10.6.2 An Efficient Algorithm for General Lagrange Elements 322
10.6.3 A Link with Mixed Finite Element Methods . . . . . 327
11 Finite Element Methods with Discontinuous Displacement,
P. Joly and C. Tsogka 331
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
11.2 Mixed Variational Formulation . . . . . . . . . . . . . . . . 332
11.3 Space Discretization . . . . . . . . . . . . . . . . . . . . . . 334
11.3.1 Choice of the Approximation Space for the Stress Tensor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
11.3.2 Choice of the Approximation Space for the Velocity . 337
11.3.3 Extension to Higher Orders and Mass Lumping . . . 339
11.4 Theoretical Issues . . . . . . . . . . . . . . . . . . . . . . . . 340
11.4.1 The Qdiv
k+1 − Pk+1 Element . . . . . . . . . . . . . . . 340
11.4.2 The Qdiv
k+1 − Qk Element . . . . . . . . . . . . . . . . 340
11.5 Time Discretization . . . . . . . . . . . . . . . . . . . . . . . 355
12 Fictitious Domains Methods for Wave Diffraction, P. Joly
and C. Tsogka 359
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
12.2 The Continuous Formulation . . . . . . . . . . . . . . . . . . 361
12.3 Finite Element Approximation and Time Discretization . . 363
12.4 Existence of the Discrete Solution and Stability . . . . . . . 367
12.5 About the Convergence Analysis . . . . . . . . . . . . . . . 370
12.5.1 FDM with the Qdiv
k+1 − Qk Element . . . . . . . . . . 370
12.5.2 FDM with the Qdiv
k+1 − Pk+1 Element . . . . . . . . . 372
12.5.3 An Abstract Result . . . . . . . . . . . . . . . . . . . 377
12.6 Illustration of the Efficiency of FDM . . . . . . . . . . . . . 382
13 Space Time Mesh Refinement Methods, G. Derveaux, P.
Joly and J. Rodr´ıguez 385
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 385
13.2 The Domain Decomposition Approach . . . . . . . . . . . . 386
13.2.1 Velocity Stress Formulation . . . . . . . . . . . . . . . 386
13.2.2 Transmission Problem . . . . . . . . . . . . . . . . . . 387
13.2.3 Variational Formulation . . . . . . . . . . . . . . . . . 388
13.3 Space Discretization . . . . . . . . . . . . . . . . . . . . . . 390
13.3.1 Semi–Discretized Variational Formulation . . . . . . . 390
13.3.2 Matrix Formulation . . . . . . . . . . . . . . . . . . . 391
13.3.3 Choice for the Approximation Spaces . . . . . . . . . 393
x
13.4 Time Discretization: the 1-2 Refinement . . . . . . . . . . . 395
13.4.1 A Conservative Time Scheme . . . . . . . . . . . . . . 395
13.4.2 Stability Analysis . . . . . . . . . . . . . . . . . . . . 396
13.4.3 Practical Computations . . . . . . . . . . . . . . . . . 399
13.5 About the Error Analysis . . . . . . . . . . . . . . . . . . . 401
13.5.1 The Case of the 1D Wave Equation . . . . . . . . . . 402
13.5.2 L2 Error Estimates . . . . . . . . . . . . . . . . . . . 404
13.5.3 Fourier Analysis Results . . . . . . . . . . . . . . . . 405
13.5.4 Numerical Illustration in the 1D Case . . . . . . . . . 407
13.6 A Post-Processed Scheme . . . . . . . . . . . . . . . . . . . 408
13.7 Generalization to Any Refinement Rate . . . . . . . . . . . 411
13.7.1 Construction of the (qc, qf) Refinement . . . . . . . . 411
13.7.2 A Post-Processed Scheme . . . . . . . . . . . . . . . . 418
13.8 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 419
14 Numerical Methods for Treating Unbounded Media, P. Joly
and C. Tsogka 425
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
14.2 Local Absorbing Boundary Condition . . . . . . . . . . . . . 427
14.2.1 The Model Problem and the Dirichlet to Neumann
Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
14.2.2 Construction of the Approximate Conditions . . . . . 431
14.2.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . 434
14.2.4 Accuracy Analysis . . . . . . . . . . . . . . . . . . . . 436
14.3 Perfectly Matched Layers . . . . . . . . . . . . . . . . . . . 438
14.3.1 Construction of the Perfectly Matched Layer for a General
Evolution Problem . . . . . . . . . . . . . . . . . 438
14.3.2 PML Model for Elastodynamics . . . . . . . . . . . . 440
14.3.3 Accuracy Analysis Using Plane Waves—Isotropic
Medium . . . . . . . . . . . . . . . . . . . . . . . . . . 442
14.3.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . 446
14.3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . 456
14.3.6 Reflection Coefficients . . . . . . . . . . . . . . . . . . 457
References 461
IV Waves in Compressible Flows 473
15 High-Order Accurate Space Discretization Methods for Computational
Fluid Dynamics, J. A. Ekaterinaris 475
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
15.2 Prior Work on High-Order Numerical Methods . . . . . . . 479
16 Governing Equations, J. A. Ekaterinaris 485
16.1 Governing Equations . . . . . . . . . . . . . . . . . . . . . . 485
xi
16.2 Conservative Form of the N-S Equations . . . . . . . . . . . 485
17 High-Order Finite-Difference Schemes, J. A. Ekaterinaris 491
17.1 High-Order Finite-Difference Schemes . . . . . . . . . . . . 491
17.2 Explicit Centered High-Order FD Schemes . . . . . . . . . . 493
17.3 Centered Compact Schemes . . . . . . . . . . . . . . . . . . 493
17.4 Boundary Closures of High-Order Schemes . . . . . . . . . . 497
17.5 Compact Schemes for the Simultaneous Evaluation of the First
and Second Derivative . . . . . . . . . . . . . . . . . . . . . 500
17.6 Modified High-Order Finite-Difference Schemes . . . . . . . 504
17.6.1 Upwind High-Order Schemes . . . . . . . . . . . . . . 505
17.7 Dispersion-Relation-Preserving (DPR) Scheme . . . . . . . . 508
17.7.1 Optimized Spatial Discretization . . . . . . . . . . . . 508
17.8 Spectral-Type Filters . . . . . . . . . . . . . . . . . . . . . . 510
17.9 Characteristic-Based Filters . . . . . . . . . . . . . . . . . . 512
17.9.1 Other Filter Formulations . . . . . . . . . . . . . . . 515
17.9.2 ENO and WENO ACM Filters . . . . . . . . . . . . . 516
17.10 Wavelet Estimation of the Sensor . . . . . . . . . . . . . . . 518
17.10.1Wavelet Analysis . . . . . . . . . . . . . . . . . . . . 518
18 ENO and WENO Schemes, J. A. Ekaterinaris 521
18.1 ENO and WENO Schemes . . . . . . . . . . . . . . . . . . . 521
18.2 High-Order Reconstruction . . . . . . . . . . . . . . . . . . . 522
18.3 Approximation in One Dimension . . . . . . . . . . . . . . . 522
18.4 One-Dimensional Conservative Approximation of the Derivative
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527
18.5 ENO Reconstruction . . . . . . . . . . . . . . . . . . . . . . 529
18.6 1D Finite-Volume ENO Scheme . . . . . . . . . . . . . . . . 532
18.7 1D Finite-Difference ENO Scheme . . . . . . . . . . . . . . . 533
18.8 WENO Approximation . . . . . . . . . . . . . . . . . . . . . 533
18.9 Application of ENO and WENO in One Dimension . . . . . 540
18.9.1 Finite-Volume Formulation . . . . . . . . . . . . . . . 541
18.9.2 Finite-Difference Formulation . . . . . . . . . . . . . 541
18.10 ENO and WENO for Characteristic Variables . . . . . . . . 542
18.11 Multidimensional ENO and WENO Reconstruction . . . . . 543
18.11.1 Finite-Volume Reconstruction for Cartesian Mesh . . 543
18.11.2 Finite-Volume ENO for the Euler Equations . . . . . 547
18.11.3Two-Dimensional Reconstruction for Triangles . . . . 548
18.11.4 Multidimensional Finite-Difference ENO . . . . . . . 551
18.12 Optimization of WENO Schemes . . . . . . . . . . . . . . . 553
18.13 Compact WENO Approximation . . . . . . . . . . . . . . . 556
18.14 Application of the Hybrid Compact-WENO Scheme for the
Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . 561
18.15 Applications of ENO and WENO . . . . . . . . . . . . . . . 562
xii
19 The Discontinuous Galerkin (DG) Method, J. A. Ekaterinaris
593
19.1 The Discontinuous Galerkin (DG) Method . . . . . . . . . . 593
19.2 Space Discretization . . . . . . . . . . . . . . . . . . . . . . 594
19.3 Triangular Element Bases . . . . . . . . . . . . . . . . . . . 596
19.3.1 First-Order Polynomials . . . . . . . . . . . . . . . . 598
19.3.2 Third-Order Polynomials . . . . . . . . . . . . . . . . 599
19.3.3 Fifth-Order Polynomials . . . . . . . . . . . . . . . . 600
19.4 Quadrilateral Element Bases . . . . . . . . . . . . . . . . . . 600
19.5 Implementation of a Quadrature-Free DG Method for Systems
of Linear Hyperbolic Equations . . . . . . . . . . . . . . . . 603
19.6 Arbitrary High Order DG Schemes . . . . . . . . . . . . . . 606
19.6.1 ADER-DG-Discretization . . . . . . . . . . . . . . . . 606
19.6.2 ADER-DG Schemes for Nonlinear Hyperbolic Systems 608
19.7 Analysis of the DG Method for Wave Propagation . . . . . 610
19.7.1 One-Dimensional Analysis . . . . . . . . . . . . . . . 611
19.7.2 Two-Dimensional Advection (Wave Equation) . . . . 612
19.8 Dissipative and Dispersive Behavior of High-Order DG Discretizations
. . . . . . . . . . . . . . . . . . . . . . . . . . . 614
19.8.1 Dispersive Properties of DG for the Advection Equation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
19.8.2 Relative Error for Fixed Mesh-Size and Large Wave
Number kh . . . . . . . . . . . . . . . . . . . . . . . . 616
19.8.3 Exponential Convergence on the Envelope 2N + 1 ≈ hk 617
19.9 Limiting of DG Expansions . . . . . . . . . . . . . . . . . . 617
19.9.1 Rectangular Elements . . . . . . . . . . . . . . . . . . 618
19.9.2 Triangular Elements . . . . . . . . . . . . . . . . . . . 619
19.10 Component-Wise Limiters . . . . . . . . . . . . . . . . . . . 622
19.11 DG Stabilization Operator . . . . . . . . . . . . . . . . . . . 624
19.11.1 Subsonic and Transonic Flow Model . . . . . . . . . . 625
19.11.2 Supersonic Flow Model . . . . . . . . . . . . . . . . . 626
19.12 DG Space Discretization of the NS Equations . . . . . . . . 626
19.12.1DG Discretization of Elliptic Problems . . . . . . . . 628
19.12.2 Approximation of the Numerical Fluxes
zh and
uh . . 630
19.12.3A DG Scheme with Compact Support . . . . . . . . . 632
19.13 The DG Variational Multiscale (VMS) Method . . . . . . . 633
19.13.1Turbulence Modeling for VMS . . . . . . . . . . . . . 640
19.14 Implicit Time Marching of DG Discretizations . . . . . . . . 642
19.15 p-Type Multigrid for DG . . . . . . . . . . . . . . . . . . . . 645
19.16 Results with the DG Method . . . . . . . . . . . . . . . . . 645
References 659
Index 681

Preface
Waves are ubiquitous in nature. We are all familiar with water waves,
sound waves, electromagnetic and seismic waves. It is fair to say that every
area of science and technology has sources of problems involving some type
of wave motion. As a result, a large arsenal of analytical techniques has been
developed to describe and analyze linear and nonlinear wave phenomena. And,
of course, the numerical simulation of wave motion, mainly the numerical
solution of the partial differential equations of wave theory, has been at the
center of attention of computational scientists since the advent of the modern
computer era.
The aim of the volume at hand is to present some modern, effective computational
methods used to describe wave propagation phenomena in selected
areas of current interest in physics and technology. One cannot hope to be
comprehensive in such an effort. It was the editors’ choice to concentrate in
four areas, which are, inevitably, close to their interests and expertise:
I. Nonlinear dispersive waves.
II. The Helmholtz equation and its paraxial approximations in underwater
acoustics.
III. Numerical methods in elastic wave propagation.
IV. Waves in compressible flows.
Each part consists of chapters (or articles) on specific topics, written by internationally
known experts. Part I begins with two articles on the simulation
of nonlinear dispersive waves from nonlinear optics. In Chapter 1, X.-P.Wang
reviews the dynamic rescaling technique and the iterative grid redistribution
method for approximating singular (blow-up) and near-singular solutions of
the Nonlinear Schr¨odinger equation. In Chapter 2, G.Fibich and S.Tsynkov
consider the Nonlinear Helmholtz equation describing time-harmonic electromagnetic
waves in Kerr media. They pose boundary-value problems for this
type of equation with nonlocal artificial boundary conditions at the nearand
far-boundaries of the half-space in the direction of which propagation
of the wave mainly takes place, and radiation conditions on the transverse
boundaries, and they solve them numerically by high-order finite difference
methods. In Chapter 3, V. A. Dougalis and D. E. Mitsotakis, after reviewing
1
2
the derivation and the well-posedness theory of a class of Boussinesq type systems
that approximate the Euler equations of water wave theory and describe
two-way propagation of long waves of small amplitude, they investigate by
analytical and numerical means (using fully discrete Galerkin-finite element
methods) solitary waves of these systems and their role in the evolution of
general solutions.
Part II consists of four chapters on computational techniques for mathematical
models of underwater sound propagation and scattering. In Chapter
4, D. A. Mitsoudis, N. A. Kampanis and V. A. Dougalis present a finite element
method for the (linear) Helmholtz equation in a general fluid waveguide
with range-dependent layer topography and concentrate on the implementation
and the coupling of the finite element method with DtN type nonlocal
boundary conditions at the inflow and outflow boundaries of the waveguide.
The next three chapters concern various issues related to paraxial (‘parabolic’)
approximations to the Helmholtz equation, that have been successfully used
to model long-range propagation of sound in the sea. First, D. J. Thomson
and G. H. Brooke in Chapter 5 present an overview of Parabolic Equation
(PE) techniques in underwater acoustics, including an introduction to
PE-based matched field processing techniques for source localization and for
decomposing the acoustic field into its modal components. In the following
Chapter 6, V. A. Dougalis, N. A. Kampanis, F. Sturm and G. E. Zouraris
address modelling and numerical issues for the PE and its higher-order wideangle
analogs in waveguides with range-dependent interfaces and bottoms in
axisymmetric and fully 3D environments, when range-dependent changes of
variable are used to make the layers horizontal. Finally, in Chapter 7, G. H.
Brooke, D. J. Thomson and the late T. W. Dawson, after reviewing various
nonlocal boundary conditions that are applied at interfaces between the computational
domain and an external half space (transverse to the direction of
propagation) in the case of the PE, they study a particular half space with linear
squared index of refraction and show how to construct nonlocal conditions
for higher-order PE’s as well.
Part III is a comprehensive introduction to modern numerical methods for
time-dependent elastic wave propagation written by P. Joly and his collaborators.
It consists of seven chapters. In the first one (Chapter 8) P. Joly provides
a general introduction and an orientation to this part of the book. In Chapter
9 he continues with a presentation of the mathematical model for elastic wave
propagation, i.e. the equations of linear elastodynamics. Chapter 10, also by
P. Joly, is a detailed exposition of full discretizations of the elastodynamics
equations using standard finite element subspaces in space. Chapter 11, by P.
Joly and C. Tsogka, concerns mixed finite element techniques, while Chapter
12, by the same authors, covers fictitious domain (finite element) methods.
In Chapter 13, G. Derveaux, P. Joly and J. Rodr´ıguez analyze space-time
mesh refinement techniques based on the principle of domain decomposition,
while, in Chapter 14, P. Joly and C. Tsogka review the state of the art of two
numerical methods for treating elastic waves in unbounded media, namely
3
the discretization of local absorbing boundary conditions and the perfectly
matched layer technique.
Part IV, by J. Ekaterinaris, is an overview of high-order, low numerical diffusion
numerical methods for complex, compressible flows of aerodynamics. It
consists of five chapters. Chapter 15 is introductory and Chapter 16 contains
the governing equations of such flows. Chapter 17 concerns high-order finite
difference schemes, Chapter 18 ENO and WENO schemes, while Chapter 19
provides an introduction of Discontinuous Galerkin methods for hyperbolic
systems.
The editors would like to express their sincere thanks to the authors of the
various chapters for contributing their work to this volume. They also wish
to express their sincere thanks to their students G. Arabatzis, I. Toulopoulos,
and S. Volanis for their help in the preparation of this volume.
Heraklion, June 2007
N. A. Kampanis
V. A. Dougalis
J. A. Ekaterinaris

楼主啊 附件呢？这么下啊 这么好的书1！！！

:11bb

[hide=100]


[/hide]

:11bb

:11bb

你跟帖够快的拉

感谢00d44管理员分享:27bb

的确是好书啊，应该下来学习一下:11bb

积分。积分啊，我可要赶快发帖赚钱了，否则很久也下不了。

感谢00d44管理员分享:11bb

:11bb
謝謝板大啦
又多一本好書
多看看多學習

的确是好书啊，应该下来学习一下:11bb

這論壇有了00d44
再怎麼樣也不孤單...
感謝分享

:27bb :27bb :27bb :27bb

谢谢啊，给你加分

够新的书籍

这么好的书，正是俺最喜欢的

最近正在做这方面的工作，比较关注 ，谢谢楼主 谢谢微网:29bb

我又杀回来接着顶贴啦O(∩_∩)O哈哈~

好像是一本论文集一样，比较散，不过有的部分可以看看

不错啊
就是电磁波跟不上
应该恶补一下了
谢谢楼主

确实是本好书！！！谢谢搂主分享！:27bb

真是好书啊，多谢谢，多谢提供者

应该是好东东，坚决支持楼主大公无私！

书很好，就是看英文老头大。看来水平有待提高。

为什么我看不到part1呢？

感谢分享，
不顶不行:21bb

也不知道自己的积分够不够啊 ，祈祷

不知道我的分够不够啊  祈祷中 不过先谢过了

:31bb ：23de ：23de

呵呵  还差几分  可惜  只好再等等了

谢谢，感谢分享！
正需要这个！

很不错的书，正要这个。。。
谢谢，感谢分享！

地热温泉热舞突然四个加快了经   感受到福建高速个

这个要看看，最好楼主先给介绍介绍

积分还是不够啊！！努力赚钱吧！谢谢楼主分享！！！

haohaohaohaohaohaohaohaohaohaohaohao

下下来学习一下，不过英语还得恶补：4de

:11bb :11bb :11bb

谢谢楼主!!!!!!!!!!!!!!!!!!!!!!!!!!!!搞了这么好的东西

:31bb :31bb :31bb

:31bb :27bb :31bb

多谢多谢，支持楼主，好人一生平安！

呵呵
楼主真会玩
把一大堆Preface放前面
我还以为要直接贴呢
呵呵
好书！收藏了！
谢谢啊！

：9de ：4de ：4de ：4de ：19de ：19de

Thank you for the sharing

谢谢分享！！！！！！！！！！！！！！！！！！！

好书啊好书
好书书

好书大家一起分享，多谢楼主！！！

积分还不够
努力发帖中 。。。

:30bb :30bb :30bb :30bb :30bb :30bb

当了士官，立即来这里
领装备

下了N多书

可是真的看了的很少很少：4de

感谢楼主，这书太好了。

感谢楼主分享，好书籍，我喜欢

谢谢楼主的分享，感谢:31bb

：9de ：17de :30bb :30bb :27bb :27bb

最近正在做这方面的工作，比较关注 ，谢谢楼主 谢谢微网

看着这么好的书，积分还是不够下载啊，着急!

楼主厉害呀，这么新的书呀，谢谢！

哎呀，积分还远远不够呀：4de

谢了，下了，:27bb ，:29bb

好人发好书！:11bb :27bb :29bb :30bb

:11bb :27bb :29bb :30bb :31bb

好资料
不顶不由人
:11bb :11bb :11bb
:27bb :27bb :27bb
:29bb :29bb :29bb

书这么新
越看越喜欢
楼主真是牛人
:27bb :27bb :27bb
:30bb :30bb :30bb

这么好的书，正是俺最喜欢的,thanks

goooooooooooooood

顶顶顶顶顶顶顶顶

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顶顶顶顶顶顶顶顶顶顶顶顶的的

好书好书！
:11bb :11bb :30bb

支持楼主，好资料，谢谢:27bb

楼主辛苦，这么新的资料。
:29bb :29bb :29bb :29bb

good,very good,thanks

good,very good,thanks

Thank you very much!!!

:29bb :29bb :29bb :29bb :29bb

好书。。。。。

的确是好书啊，应该下来学习一下

谢谢分享
谢谢楼主!

自己顶一贴

:27bb 5# 00d44

谢谢谢谢谢谢

谢谢啊，给你加分

：18de:13bbgood!

加分加分，太应该了

下载学习下

thank you very much!!!!!!!!!!!!!!!!!!!!!

终于可以发言了，呵呵。

我看不到Part 1，只好Google一下了。结果，嘿嘿，不需要积分（但要等待60秒）。
http://depositfiles.com/en/files/c722ingf3

感谢楼主提供了这样一本不错的书籍，个别章节确能为自己所用。

好书大家分享

可惜权限不够

确实得需要赚钱了啊！

我靠，我才9，啥时候能凑到100啊？

Thank you for your sharing

{:6_944:}{:6_923:}

下之，哈哈，不错

连这种书都可以找到，厉害。

支持新书！！！！！！！！

好书，学习学习哈

This is a basic theory books !

的确是好书啊

积分还不够，急！！！

楼主，门槛设的太高了，需要100积分呀，我还得赚积分。

终于下载到了，感谢微波网

感谢啊

谢谢分享。。。。。。。。。。。。。。。

谢谢分享。。。。。。。。。

谢谢分享。。。。。。。。。

thanks for sharing!

sdsakjdsafafhaskj

希望能有權利下載來看
感謝分享