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Mathematical Aspects of Numerical Solution of Hyperbolic Systems : Monographs and Surveys in Pure and Applied Mathematics - A. G. Kulikovsky

Mathematical Aspects of Numerical Solution of Hyperbolic Systems

Monographs and Surveys in Pure and Applied Mathematics

Hardcover Published: 21st December 2000
ISBN: 9780849306082
Number Of Pages: 560

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This important new book sets forth a comprehensive description of various mathematical aspects of problems originating in numerical solution of hyperbolic systems of partial differential equations. The authors present the material in the context of the important mechanical applications of such systems, including the Euler equations of gas dynamics, magnetohydrodynamics (MHD), shallow water, and solid dynamics equations. This treatment provides-for the first time in book form-a collection of recipes for applying higher-order non-oscillatory shock-capturing schemes to MHD modelling of physical phenomena.
The authors also address a number of original "nonclassical" problems, such as shock wave propagation in rods and composite materials, ionization fronts in plasma, and electromagnetic shock waves in magnets. They show that if a small-scale, higher-order mathematical model results in oscillations of the discontinuity structure, the variety of admissible discontinuities can exhibit disperse behavior, including some with additional boundary conditions that do not follow from the hyperbolic conservation laws. Nonclassical problems are accompanied by a multiple nonuniqueness of solutions. The authors formulate several selection rules, which in some cases easily allow a correct, physically realizable choice.
This work systematizes methods for overcoming the difficulties inherent in the solution of hyperbolic systems. Its unique focus on applications, both traditional and new, makes Mathematical Aspects of Numerical Solution of Hyperbolic Systems particularly valuable not only to those interested the development of numerical methods, but to physicists and engineers who strive to solve increasingly complicated nonlinear equations.

Industry Reviews

"The book is a substantial addition to the existing literature It will be of interest to students and researchers in fluid dynamics and continuum mechanics in various field of physics." -European Mathematical Society Newsletter, No. 41 (September 2001)" this bookis as a sort of encyclopedia on numerical techniques applied to hyperbolic systems. Being free of, although important, mathematical and physical details, it allows the authors to focus the reader's attention on the core of numerics. The book is worthy of being in the library of everyone interested not only in numerical methods, but also in applied mathematics, mechanics, physics, and engineering, since the hyperbolic conservation laws are the basis of these areas of research." -Applied Mathematics Review, Vol. 55, no. 3, May 2002

Hyperbolic Systems of Partial Differential Equationsp. 1
Quasilinear systemsp. 1
Hyperbolic systems of quasilinear differential equationsp. 2
Definitionsp. 2
Systems of conservation lawsp. 4
Mechanical examplesp. 5
Nonstationary equations of gas dynamicsp. 5
Stationary Euler equationsp. 8
Shallow water equationsp. 11
Equations of ideal magnetohydrodynamicsp. 12
Elasticity equationsp. 15
Properties of solutionsp. 17
Classical solutionsp. 17
Generalized solutionsp. 21
Small-amplitude shocksp. 25
Evolutionary conditions for shocksp. 27
Entropy behavior on discontinuitiesp. 29
Disintegration of a small arbitrary discontinuityp. 31
Numerical Solution of Quasilinear Hyperbolic Systemsp. 33
Introductionp. 33
Methods based on the exact solution of the Riemann problemp. 37
The Godunov method of the first orderp. 38
Exact solution of the Riemann problemp. 40
Methods based on approximate Riemann problem solversp. 43
Courant-Isaacson-Rees-type methodsp. 43
Roe's schemep. 55
The Osher numerical schemep. 58
Generalized Riemann problemp. 65
The Godunov method of the second orderp. 67
Multidimensional schemes and their stability conditionsp. 72
Reconstruction procedures and slope limitersp. 76
Preliminary remarksp. 76
TVD schemesp. 78
Monotone and limiting reconstructionsp. 80
Genuine TVD and TVD limiting reconstructionsp. 87
TVD limiters of nonsymmetric stencilp. 92
Multidimensional reconstructionp. 94
Boundary conditions for hyperbolic systemsp. 101
General notionsp. 101
Nonreflecting boundary conditionsp. 102
Evolutionary boundary conditionsp. 107
Shock-fitting methodsp. 109
Floating shock fittingp. 109
Shock fitting on moving gridsp. 113
Entropy correction proceduresp. 115
Final remarksp. 119
Gas Dynamic Equationsp. 121
Systems of governing equationsp. 121
Two-temperature gas dynamic equationsp. 124
The mixture of ideal gases in chemical nonequilibriump. 127
The Godunov method for gas dynamic equationsp. 129
Exact solution of the Riemann problemp. 132
Elementary solution 1: Shock wavep. 132
Elementary solution 2: Contact discontinuityp. 136
Elementary solution 3: Rarefaction wavep. 136
General exact solutionp. 139
An arbitrary EOSp. 147
Approximate Riemann problem solversp. 151
The Courant-Isaacson-Rees method for an arbitrary EOSp. 152
Computation of shock-induced phenomena by the CIR methodp. 154
The CIR-simulation of jet-like structures in laser plasmap. 158
Roe's methodp. 163
Roe's Riemann problem solver for an arbitrary EOSp. 169
Osher-Solomon numerical schemep. 171
Shock-fitting methodsp. 175
Discontinuities as boundaries of the computational regionp. 175
Floating shock-fitting proceduresp. 186
Shock-fitting on moving gridsp. 189
Self-adjusting gridsp. 190
Stationary gas dynamicsp. 197
Systems of governing equationsp. 197
The Godunov method. The CIR and Roe's schemesp. 201
Exact solution of the Riemann problemp. 203
General exact solutionp. 212
Solar wind - interstellar medium interactionp. 213
Physical formulation of the problemp. 213
Nonreflecting boundary conditionsp. 218
Numerical resultsp. 221
A note on Godunov-type methods for relativistic hydrodynamicsp. 224
Shallow Water Equationsp. 225
System of governing equationsp. 225
The Godunov method for shallow water equationsp. 228
Exact solution of the Riemann problemp. 231
Elementary solution 1: Hydraulic jumpp. 231
Elementary solution 2: Tangential discontinuityp. 235
Elementary solution 3: Riemann wavep. 235
General exact solutionp. 237
Results of numerical analysisp. 245
Approximate Riemann problem solversp. 257
The CIR methodp. 257
Roe's methodp. 258
The Osher-Solomon solverp. 261
Stationary shallow water equationsp. 262
System of governing equationsp. 263
The Godunov method. The CIR and Roe's schemesp. 265
Exact solution of the Riemann problemp. 266
General exact solutionp. 275
Magnetohydrodynamic Equationsp. 277
MHD system in the conservation-law formp. 277
Classification of MHD discontinuitiesp. 285
Evolutionary MHD shocksp. 288
Evolutionary diagramp. 288
Convenient relations on MHD shocksp. 290
Evolutionarity of perpendicular, parallel, and singular shocksp. 291
Jouget pointsp. 294
High-resolution numerical schemes for MHD equationsp. 295
The Osher-type methodp. 296
Piecewise-parabolic methodp. 297
Roe's characteristic decomposition methodp. 298
Numerical tests with the Roe-type schemep. 305
Modified MHD systemp. 323
Shock-capturing approach and nonevolutionary solutions in MHDp. 328
Preliminary remarksp. 328
Simplified MHD equations and related discontinuitiesp. 332
Shock structure in solutions of the simplified systemp. 333
Nonstationary processes in the structure of nonevolutionary shock wavesp. 335
Numerical experiments based on the full set of MHD equationsp. 337
Numerical disintegration of a compound wavep. 339
Strong background magnetic fieldp. 345
Elimination of numerical magnetic chargep. 348
Preliminary remarksp. 348
Application of the vector potentialp. 349
The use of an artificial scalar potentialp. 350
Application of the modified MHD systemp. 351
Application of staggered gridsp. 352
Solar wind interaction with the magnetized interstellar mediump. 356
Statement of the problemp. 357
Numerical algorithmp. 359
Numerical results: axisymmetric casep. 363
Numerical results: rotationally perturbed flowp. 368
A note on the MHD flow over an infinitely conducting cylinderp. 372
Numerical results: three-dimensional modellingp. 374
Solid Dynamics Equationsp. 379
System of governing equationsp. 379
Solid dynamics with an arbitrary EOSp. 380
Conservative form of elastoviscoplastic solid dynamicsp. 392
Dynamics of thin shellsp. 396
CIR method for the calculation of solid dynamics problemsp. 399
Numerical simulation of spallation phenomenap. 404
CIR method for studying the dynamics of thin shellsp. 410
The Klein-Gordon equationp. 415
Dynamics equations of cylindrical shellsp. 416
Dynamics equations of orthotropic shellsp. 418
Selection of rapidly oscillating componentsp. 419
Nonclassical Discontinuities and Solutions of Hyperbolic Systemsp. 423
Evolutionary conditions in nonclassical casesp. 425
Structure of fronts. Additional boundary conditions on the frontsp. 427
Equations describing the discontinuity structurep. 429
Formulation of the structure problem and additional assumptionsp. 431
Behavior of the solution as [xi] [right arrow] [plus or minus infinity]p. 432
Additional relations on discontinuitiesp. 434
Main result and its discussionp. 435
A remark on deriving additional relations when condition (7.2.7) is not satisfiedp. 436
Hugoniot manifoldp. 438
Behavior of the Hugoniot curve in the vicinity of Jouget points and nonuniqueness of solutions of self-similar problemsp. 439
Nonlinear small-amplitude waves in anisotropic elastic mediap. 447
Basic equationsp. 447
Quasilongitudinal wavesp. 449
Quasitransverse wavesp. 450
Riemann wavesp. 451
Shock wavesp. 452
Self-similar problems and nonuniqueness of solutionsp. 455
Waves in viscoelastic media, vanishing viscosityp. 456
Role of the wave anisotropy and passage to the limit g [right arrow] 0p. 459
Final conclusionsp. 460
Electromagnetic shock waves in ferromagnetsp. 461
Long-wave approximation. Elastic analogyp. 461
Structure of electromagnetic shock wavesp. 464
The set of admissible discontinuitiesp. 470
Nonuniqueness of solutionsp. 470
Shock waves in composite materialsp. 472
Basic equations and the discontinuity structurep. 472
Discontinuity structure; admissible discontinuitiesp. 474
Case h ] 0p. 474
Case h [ 0p. 478
Longitudinal nonlinear waves in elastic rodsp. 479
Large-scale modelp. 479
Model for moderate-scale motionsp. 481
Equations describing the discontinuity structurep. 482
Admissible discontinuitiesp. 483
More precise large-scale model. Nonuniquenessp. 486
Ionization fronts in a magnetic fieldp. 487
Large-scale modelp. 487
Moderate-scale modelp. 488
The set of admissible discontinuitiesp. 490
The simplest self-similar problemp. 494
Variation of the gas velocity across ionization frontsp. 495
Constructing the solution of the piston problemp. 500
Discussionp. 501
Bibliographyp. 503
Indexp. 535
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780849306082
ISBN-10: 0849306086
Series: Monographs and Surveys in Pure and Applied Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 560
Published: 21st December 2000
Publisher: CHAPMAN & HALL
Country of Publication: US
Dimensions (cm): 24.28 x 16.54  x 3.35
Weight (kg): 0.9
Edition Number: 1

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