Handbook of numerical analysis: Volume 17 Handbook of numerical methods for hyperbolic problems : basic and fundamental issues
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| Format: | Buch |
| Sprache: | Englisch |
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Amsterdam
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[2016]
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| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029357731&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xxiii, 641 Seiten Illustrationen, Diagramme (teilweise farbig) |
| ISBN: | 9780444637895 |
Internformat
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| 245 | 1 | 0 | |a Handbook of numerical analysis |n Volume 17 |p Handbook of numerical methods for hyperbolic problems : basic and fundamental issues |c general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| 264 | 1 | |a Amsterdam |b North-Holland |c [2016] | |
| 300 | |a xxiii, 641 Seiten |b Illustrationen, Diagramme (teilweise farbig) | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 700 | 1 | |a Abgrall, Rémi |e Sonstige |0 (DE-588)1052897819 |4 oth | |
| 700 | 1 | |a Ciarlet, Philippe G. |d 1938- |e Sonstige |0 (DE-588)143368362 |4 oth | |
| 700 | 1 | |a Shu, Chi-Wang |d 1957- |e Sonstige |0 (DE-588)1018436952 |4 oth | |
| 700 | 1 | |a Lions, Jacques-Louis |d 1928-2001 |e Sonstige |0 (DE-588)124055397 |4 oth | |
| 700 | 1 | |a Du, Qiang |d 1964- |0 (DE-588)1188249320 |4 edt | |
| 773 | 0 | 8 | |w (DE-604)BV002745459 |g 17 |
| 856 | 4 | 2 | |m Digitalisierung UB Passau - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029357731&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-029357731 | |
Datensatz im Suchindex
| _version_ | 1819360626290458624 |
|---|---|
| adam_text | Contents
Contributors xvii
Introduction xxi
1. Introduction to the Theory of Hyperbolic
Conservation Laws 1
C.M. Dafermos
1 Introduction 1
2 Basic Structure of Hyperbolic Conservation Laws 2
3 Strictly Hyperbolic Systems in One Spatial Dimension 10
References 18
2. The Riemann Problem: Solvers and Numerical Fluxes 19
E.F Toro
1 Preliminaries 20
1.1 Definitions and Simple Examples 20
1 .2 Hyperbolic Systems and Finite Volume Methods 23
2 Exact Solution of the Riemann Problem for the
Euler Equations 27
2.1 Equations and Structure of the Solution 27
2.2 Pressure and Velocity in the Star Region 29
2.3 The Complete Solution and the 3D Case 32
2.4 Uses of the Exact Solution of the Riemann
Problem 33
2.5 Approximate Riemann Solvers: Beware 35
3 The Roe Approximate Riemann Solver 36
4 The HLL Approximate Riemann Solver 37
5 The HLLC Approximate Riemann Solver 40
5.1 Derivation of the HLLC Flux 40
5.2 Wave Speed Estimates for HLL and HLLC 42
6 A Numerical Version of the Osher-Solomon
Riemann Solver 43
7 Other Approaches to Constructing Numerical Fluxes 45
8 Concluding Remarks 48
Acknowledgements 49
References 49
v
vi
Contents
3. Classical Finite Volume Methods 55
T. Sonar
1 Some Philosophical Remarks 55
2 On the Lax-Wendroff Theorem 58
3 Historical Remarks 58
4 Weak Solutions and Finite Volume Methods 61
5 The Cell-Centred Scheme of Jameson, Schmidt and Turkel 63
6 Cell-Vertex Schemes on Quadrilateral Grids 67
7 Finite Volume Methods on Unstructured Grids 69
7.1 Cell-Centred Finite Volume Methods 69
7.2 Vertex-Centred Finite Volume Methods 72
7.3 Remarks on Recovery 74
References 75
4. Sharpening Methods for Finite Volume Schemes 77
B. Despres, S. Kokh, and F. Lagoutiere
1 Introduction 78
2 Sharpening Methods for Linear Equations 78
2.1 High-Order Methods 79
2.2 Compression Within a BV Setting 61
2.3 inequality and Antidiffusion 64
2.4 Glimm s Method 66
2.5 PDE Models and Sharpening Methods 67
2.6 Nature of the Grid/Mesh 68
2.7 Interface Reconstruction and VOF 69
2.8 Vofire 69
3 Coupling With Hyperbolic Nonlinear Equations 94
3.1 An Example of Discretization for Compressible
Flows With Two Components Separated by a Sharp
Interface 94
3.2 Example of Other Evolution Equation Involving
Sharp Interfaces 97
3.3 Cut-Cells and CFL Condition 98
References 98
5. ENO and WENO Schemes 103
K-T. Zhang and C.-W. Shu
1 Introduction 104
2 ENO and WENO Approximations 105
2.1 Reconstruction 105
2.2 ENO Approximation 107
2.3 WENO Approximation 108
3 ENO and WENO Schemes for Hyperbolic
Conservation Laws 110
3.1 Finite Volume Schemes 110
Contents
VII
3.2 Finite Difference Schemes 111
3.3 Remarks on Multidimensional Problems and Systems 11 2
4 Selected Topics of Recent Developments 113
4.1 Unstructured Meshes 113
4.2 Steady State Problems 11 7
4.3 Time Discretizations for Convection—Diffusion Problems 118
4.4 Accuracy Enhancement 119
Acknowledgements 11 9
References 120
6. Stability Properties of the ENO Method 1 23
(J. S. Fjordholm
1 Introduction 123
2 The ENO Reconstruction Method 125
2.1 Choosing the Stencil Index 126
3 Application to Conservation Laws 128
3.1 Finite Volume Methods 128
3.2 TVD ENO Schemes 129
3.3 Convergence of High-Order Schemes 130
4 ENO Stability Properties 133
4.1 Immediate Properties 133
4.2 The Sign Property 134
4.3 Upper Bound on Jumps 136
4.4 The ENO TV Conjecture 1 36
4.5 Mesh-Dependent Properties 138
4.6 ENO Deficiencies 142
5 Summary 143
Acknowledgements 144
References 144
7. Stability, Error Estimate and Limiters
of Discontinuous Galerkin Methods 147
J. Qiu and Q. Zhang
1 Introduction 148
2 Implementation of DG Methods 149
2.1 Semidiscrete Version 150
2.2 SSPRK Algorithms 151
2.3 Limiters 152
3 Stability 152
3.1 Linear Stability in ¿2-Norm 153
3.2 Nonlinear Stability 156
4 Error Estimates 157
4.1 Scalar Equation with Smooth Solution 157
4.2 Symmetrizabie System with Smooth Solution 1 58
4.3 Scalar Equation with Discontinuous Initial Solution 159
4.4 Other Error Estimates 160
Contents
• • •
VIII
5 Limiters for Discontinuous Gaierkin Methods 160
5.1 Traditional Limiters 162
5.2 WENO Reconstruction as a Limiter for the RKDG Method 163
5.3 Hermite WENO Reconstruction 165
5.4 A Simple WENO-Type Limiter 166
5.5 A Simple and Compact HWENO Limiter 167
6 Concluding and Remarks 168
References 168
8. HDG Methods for Hyperbolic Problems 173
B. Cockburn, N.C. Nguyen, and J. Peraire
1 Introduction 174
2 The Acoustics Wave Equation 1 74
2.1 Spatial Discretization 175
2.2 Temporal Discretization 177
2.3 SSP-RK Methods 1 79
2.4 Postprocessing 180
2.5 Numerical Results 181
3 The Elastic Wave Equations 181
3.1 Spatial Discretization 184
3.2 Local Postprocessing 186
3.3 Numerical Results 186
4 The Electromagnetic Wave Equations 189
4.1 Numerical Discretization 189
4.2 Local Postprocessing 190
4.3 Numerical Results 191
5 Bibliographic Notes 191
5.1 Time-Dependent Wave Propagation 191
5.2 Time-Harmonic Wave Propagation 194
5.3 Further Reading Material 195
Acknowledgements 195
References 195
9. Spectral Volume and Spectral Difference
Methods 199
Z.J. Wang, Y. Liu, C. Lacor, and J.L.F. Azevedo
1 Introduction 200
2 One-Dimensional Formulations 203
2.1 SV Method 203
2.2 SD Method 205
2.3 Equivalence of the SV and SD Methods and Their
Stability 206
3 Two-Dimensional Formulation on the Simplex 207
3.1 SV Method 208
3.2 SD Method 210
3.3 Efficiency and Stability 214
Contents
IX
4 Numerical Examples 215
4.1 Double Mach Reflection 215
4.2 Rayleigh—Taylor Instability Problem With
Solution-Based Grid Adaptation 217
4.3 Aerodynamic Performance of Flapping Wing 219
5 Conclusions 221
Acknowledgements 221
References 222
10. High-Order Flux Reconstruction Schemes 227
F.D. Witherden, RE. Vincentand A. Jameson
1 Introduction 228
2 FRinID 230
2.1 Advection Problems 230
2.2 Advection Diffusion 233
3 FR in Multidimensions 235
3.1 Overview 235
3.2 Tensor Product Elements 235
3.3 Simplex Elements 236
4 Stability and Accuracy of FR Schemes 241
4.1 Energy Stability 241
4.2 von Neumann Analysis 243
4.3 Nonlinear Stability 243
5 Implementation 244
5.1 Overview 244
5.2 Salient Aspects of an FR Implementation 245
6 Applications 246
6.1 Solving the Euler and Navier—Stokes
Equations 246
6.2 Flow Over a Circular Cylinder 247
6.3 Flow Over an SD7003 Wing 253
6.4 T106c Low-Pressure Turbine Cascade 255
7 Summary 258
References 260
11. Linear Stabilization for First-Order PDEs 265
A. Ern and J.-L. Guermond
1 Friedrichs Systems 266
1.1 Basic Ideas and Model Problem 266
1.2 Example 1: Advection—Reaction Equation 267
1.3 Example 2: Maxwell s Equations 268
2 Weak Formulation and Weli-Posedness for Friedrichs
Systems 269
2.1 The Graph Space 269
2.2 The Boundary Operators 270
2.3 Well-Posedness 271
X
Contents
3 Residual-Based Stabilization 273
3.1 Least-Squares Formulation 273
3.2 Least-Squares Approximation Using Finite Elements 274
3.3 Galerkin/Least-Squares 275
4 Boundary Penalty for Friedrichs Systems 277
4.1 Model Problem 278
4.2 Boundary Penalty Method 278
4.3 Galerkin Least-Squares Stabilization with Boundary Penalty 280
5 Fluctuation-Based Stabilization 280
5.1 Abstract Theory for Fluctuation-Based Stabilization 281
5.2 Continuous Interior Penalty 283
5.3 Two-Scale Stabilization, Local Projection and Subgrid
Viscosity 284
References 287
12. Least-Squares Methods for Hyperbolic Problems 289
R Bochev and M. Gunzburger
1 Introduction 290
2 LSFEM for Hyperbolic Problems 292
3 Conservation Laws 293
4 Energy Balances 294
4.1 Energy Balances in Hilbert Spaces 294
4.2 Energy Balances in Banach Spaces 296
5 Continuous Least-Squares Principles 296
5.1 Extension to Time-Dependent Conservation Laws 298
6 LSFEM in a Hilbert Space Setting 299
6.1 Conforming LSFEMs 299
6.2 Nonconforming Methods 300
7 Residual Minimization Methods in a Banach Space Setting 302
7.1 An L?{Q) Minimization Method 302
7.2 Regularized L1(D) Minimization Method 303
8 LSFEMs Based on Adaptively Weighted ¿2(i!) Norms 305
8.1 An Iteratively Reweighted LSFEM 305
8.2 A Feedback LSFEM 306
9 Examples 308
9.1 Approximation of Smooth Solutions 308
9.2 Approximation of Discontinuous Solutions 309
10 A Summary of Conclusions and Recommendations 314
Acknowledgements 315
References 315
13. Staggered and Colocated Finite Volume
Schemes for Lagrangian Hydrodynamics 319
R. Loubbre, P.-H. Maire, and B. Rebourcet
1 Historical Background on Lagrangian Computational
Fluid Dynamics 320
Contents
xi
2 Lagrangian Hydrodynamics 324
2.1 Physical Conservation Laws Written Under Integral Form 324
2.2 Thermodynamic Closure 325
2.3 Physical Conservation Laws Written Under Local Form 326
2.4 Geometrical Conservation Law 327
3 GCL and Related Discrete Operators 327
3.1 Grid Notation and Assumptions 327
3.2 Compatible Discretization of the GCL 328
3.3 Discrete Divergence and Gradient Operators 330
3.4 Hourglass Fixes 332
4 Discrete Compatible Staggered Lagrangian
Hydrodynamics — SG H 334
4.1 Notation and Assumptions 334
4.2 Semidiscrete Compatible Discretization of the GCL 334
4.3 Semidiscrete Momentum Equation on the Dual
Cell cop 335
4.4 Semidiscrete Internal Energy Equation on the
Primal Cell coc 336
4.5 Compatible Discretization of Additional Subcell Forces 337
4.6 Time Discretization 341
5 Discrete Colocated Lagrangian Hydrodynamics —CLH 342
5.1 Notation and Assumptions 342
5.2 Subcell Force-Based Discretization 343
5.3 Local Entropy Inequality 343
5.4 Conservation of Total Energy and Momentum 344
5.5 Nodal Solver 345
5.6 First-Order Time Discretization 346
5.7 Second-Order Extension 347
Acknowledgements 348
References 348
14. High-Order Mass-Conservative Semi-Lagrangian
Methods for Transport Problems 353
J.-M.Qiu
1 Introduction 354
2 Mass-Conservative SL Schemes 357
2.1 SL Finite Difference WENO Scheme 358
2.2 Mass-Conservative SL DG Scheme 363
3 Standard Test Sets 367
3.1 1-D Problems 367
3.2 2-D Linear Passive Advection Problems 369
4 Nonlinear Vlasov-SL DG and Incompressible Euler System 374
4.1 Vlasov—Poisson Simulations 374
4.2 Guiding Center Model for a Kelvin—Helmholtz Instability 377
4.3 2-D incompressible Euler (Bell et al., 1989) 378
Acknowledgements 380
References 380
384
385
385
388
390
393
394
394
395
396
399
399
399
103
404
404
406
409
415
416
419
421
421
426
434
435
441
442
444
445
447
448
448
449
Front-Tracking Methods
D. She, R. Kaufman, H. Lim, ]. Melvin, A. Hsu,
and J. Glimm
1 Introduction
2 FT as a Numerical Algorithm
2.1 FTI Overview
2.2 Application Specific (Client) Algorithms,
Nonconservative Tracking
2.3 Client Algorithms, Conservative Tracking
2.4 Geometric (FTI) Algorithms
3 Scientific Uses of FT
3.1 Benchmark Problems
3.2 Verification and Validation Examples
3.3 A Complex Physics Example
4 Conclusions
Acknowledgements
References
Moretti s Shock-Fitting Methods on Structured
and Unstructured Meshes
A. Bonfiglioli, R. Paciorri, F Nasutl, and M. Onofri
1 Introduction
2 Shock-Fitting, Upwinding and Modern Shock-Capturing
Schemes
3 Boundary Shock-Fitting
4 Floating Shock-Fitting
4.1 Floating Shock-Fitting Results
4.2 Viscous Flows
4.3 Complex Flows
5 Shock-Fitting for Unstructured Grids
5.1 Unstructured Shock-Fitting: Algorithmic Features
5.2 Unstructured Shock-Fitting: Applications
6 Conclusions
References
Spectral Methods for Hyperbolic Problems1
J.S. Hesthaven
1 Introduction
2 The Spectral Expansion
2.1 Smooth Problems
2.2 Nonsmooth Problems
2.3 The Duality Between Modes and Nodes
3 Spectral Methods
3.1 Galerkin Methods
Contents
XIII
3.2 Collocation Methods 450
3.3 Interlude on Polynomial Methods and Boundary
Conditions 452
4 Stability and Convergence of Nonlinear Problems 455
4.1 Skew-Symmetric Form 455
4.2 Filtering for Stability 456
4.3 Vanishing Viscosity Techniques 458
5 Postprocessing Techniques 459
5.1 Filtering for Accuracy 460
5.2 Gegenbauer Reconstruction 461
References 463
18. Entropy Stable Schemes 467
E. Tadmor
1 Entropie Systems of Conservation Laws 468
1.1 Entropy Pairs 469
1.2 Entropy Inequality 470
1.3 The One-Dimensional Setup 470
2 Discrete Approximations and Entropy Stability 471
2.1 Examples 472
2.2 Entropy Stability 473
3 Entropy Stable Schemes for Scalar
Conservation Laws 473
3.1 Monotone Schemes 473
3.2 E-Schemes 475
3.3 Numerical Viscosity I 475
4 Semidiscrete Schemes for Systems of
Conservation Laws 477
4.1 Entropy Variables 477
4.2 Entropy Conservative Fluxes 478
4.3 Flow Much Numerical Viscosity 479
4.4 Scalar Entropy Stability Revisited 479
4.5 Numerical Viscosity II 481
4.6 Entropy Conservative Fluxes—Systems of
Conservation Laws 482
5 Fully Discrete Schemes for Systems of
Conservation Laws 484
5.1 Numerical Viscosity III 485
5.2 A Flomotopy Method 487
6 Higher-Order Methods 487
7 Multidimensional Systems of Conservation Laws 488
7.1 Cartesian Grids 488
7.2 Unstructured Grids 489
Acknowledgements 490
References 490
498
499
500
500
501
502
502
503
503
505
505
507
510
51 1
512
512
513
515
516
517
518
521
522
525
526
527
529
533
534
534
535
536
537
542
Entropy Stable Summation-by-Parts Formulations
for Compressible Computational Fluid Dynamics
M.H. Carpenter, T.C. Fisher, EJ. Nielsen, M. Parsani,
M. Svard, and N. Yamaleev
1 introduction
2 The Compressible NSE
2.1 Governing Equations
2.2 Continuous Entropy Analysis
3 SBP Operators
3.1 Mimetic Operators
3.2 Complementary Grid and Telescopic Flux Form
3.3 Extension to Multiple Dimensions
3.4 Diagonal-Norm SBP Operators
3.5 The Semidiscrete Operators WitFi Boundary
and Interface Conditions
4 Semidiscrete and Fully Discrete Entropy Analysis
4.1 Fully Discrete Operators
5 Entropy Stable interior Interface Coupling
6 Entropy Stable Solid Wall Boundary Conditions
7 Entropy Stable WENO Formulations
7.1 An Entropy Comparison Approach
8 Conservation of Entropy in Curvilinear Coordinates
8.1 Coordinate Transformations and Geometric
Conservation Laws
8.2 Curvilinear Conservation and Stability
9 Results: Accuracy and Robustness
9.1 Taylor—Green Vortex
9.2 Computation of a Square Cylinder in Supersonic
Free Stream
9.3 Supersonic Cylinder
10 Conclusions
References
Central Schemes: A Powerful Black-Box
Solver for Nonlinear Hyperbolic PDEs
A. Kurganov
1 AVery Brief Theoretical Background
2 Finite-Volume Framework
3 First-Order Upwind Schemes
4 First-Order Central Schemes
5 High-Order Finite-Volume Methods
5.1 Second-Order Upwind Schemes
5.2 Second-Order Nessyahu—Tadmor Scheme
5.3 FHigh-Order Schemes
6 Central-Upwind Schemes
6.1 Semidiscrete Central-Upwind Schemes
Contents xv
Acknowledgements 544
References 544
21. Time Discretization Techniques 549
S. Gottlieb and DJ. Ketcheson
1 Overview 550
2 Classical Methods 552
2.1 Runge—Kutta 553
2.2 Multistep 554
2.3 Multistage Multistep Methods 555
2.4 Taylor Series Methods 555
2.5 Multistage Multiderivative Methods 555
3 Deferred Correction Methods 557
4 Strong Stability Preserving Methods 558
4.1 The SSP Property 559
4.2 Optimal Explicit Methods 560
4.3 Optimal Implicit Methods 563
4.4 Optimal SSP Runge—Kutta Methods for Linear
Constant Coefficient Problems 564
4.5 Optimal Multistep Runge—Kutta Methods 565
4.6 Strong Stability Properties of Multiderivative
Methods 567
4.7 Widespread Applicability of SSP Methods 568
5 Other Numerically Optimized Methods 569
6 IMEX Methods 570
7 Exponential Time Differencing 572
8 Multirate Time Stepping 574
9 Parallel in Time Methods 575
9.1 Concurrency Across the Method 575
9.2 Concurrency Across the Time Domain 575
References 576
22. The Fast Sweeping Method for Stationary
Hamilton-Jacobi Equations 585
H.Zhao
1 Introduction to Hamilton-Jacobi Equation 586
2 Survey of Numerical Methods for Hamilton-Jacobi
Equations 588
3 The FSM 591
3.1 The FSM on a Rectangular Grid 591
3.2 The FSM for General Convex Hamilton-Jacobi
Equations and on Triangular Meshes 593
3.3 Extension of the FSM 596
Acknowledgement 599
References 599
xvi
Contents
23. Numerical Methods for Hamilton—Jacobi
Type Equations 603
M. Falcone and R. Ferretti
1 Introduction and Motivations 604
1.1 Front Propagation via Level Set Method 604
1.2 The Infinite Horizon Problem 605
2 Basics on Viscosity Solutions 605
2.1 Convergence Results 609
3 Evolutive Problems 612
3.1 Monotone Schemes in Differenced Form 612
3.2 SL Discretization 615
3.3 Convergence 617
4 Stationary Problems 619
4.1 Discretization in Differenced Form 619
4.2 SL Discretization 620
4.3 Convergence and a Priori Error Estimates 620
5 High-order Approximation Methods 621
5.1 Theoretical Tools 621
5.2 High-order FD Schemes 622
5.3 High-order SL Schemes 623
5.4 Discontinuous Galerkin 623
5.5 Filtered Schemes 624
References 625
Index 627
|
| any_adam_object | 1 |
| author2 | Du, Qiang 1964- |
| author2_role | edt |
| author2_variant | q d qd |
| author_GND | (DE-588)1052897819 (DE-588)143368362 (DE-588)1018436952 (DE-588)124055397 (DE-588)1188249320 |
| author_facet | Du, Qiang 1964- |
| building | Verbundindex |
| bvnumber | BV043948829 |
| classification_rvk | SK 900 |
| ctrlnum | (OCoLC)967907926 (DE-599)BVBBV043948829 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV043948829 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T17:49:31Z |
| institution | BVB |
| isbn | 9780444637895 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-029357731 |
| oclc_num | 967907926 |
| open_access_boolean | |
| owner | DE-384 DE-739 DE-11 |
| owner_facet | DE-384 DE-739 DE-11 |
| physical | xxiii, 641 Seiten Illustrationen, Diagramme (teilweise farbig) |
| publishDate | 2016 |
| publishDateSearch | 2016 |
| publishDateSort | 2016 |
| publisher | North-Holland |
| record_format | marc |
| spellingShingle | Handbook of numerical analysis |
| title | Handbook of numerical analysis |
| title_auth | Handbook of numerical analysis |
| title_exact_search | Handbook of numerical analysis |
| title_full | Handbook of numerical analysis Volume 17 Handbook of numerical methods for hyperbolic problems : basic and fundamental issues general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_fullStr | Handbook of numerical analysis Volume 17 Handbook of numerical methods for hyperbolic problems : basic and fundamental issues general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_full_unstemmed | Handbook of numerical analysis Volume 17 Handbook of numerical methods for hyperbolic problems : basic and fundamental issues general editor: P. G. Ciarlet (Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie), J. L. Lions |
| title_short | Handbook of numerical analysis |
| title_sort | handbook of numerical analysis handbook of numerical methods for hyperbolic problems basic and fundamental issues |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029357731&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002745459 |
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