Hyperbolic Systems of Partial Differential Equations | p. 1 |
Quasilinear systems | p. 1 |
Hyperbolic systems of quasilinear differential equations | p. 2 |
Definitions | p. 2 |
Systems of conservation laws | p. 4 |
Mechanical examples | p. 5 |
Nonstationary equations of gas dynamics | p. 5 |
Stationary Euler equations | p. 8 |
Shallow water equations | p. 11 |
Equations of ideal magnetohydrodynamics | p. 12 |
Elasticity equations | p. 15 |
Properties of solutions | p. 17 |
Classical solutions | p. 17 |
Generalized solutions | p. 21 |
Small-amplitude shocks | p. 25 |
Evolutionary conditions for shocks | p. 27 |
Entropy behavior on discontinuities | p. 29 |
Disintegration of a small arbitrary discontinuity | p. 31 |
Numerical Solution of Quasilinear Hyperbolic Systems | p. 33 |
Introduction | p. 33 |
Methods based on the exact solution of the Riemann problem | p. 37 |
The Godunov method of the first order | p. 38 |
Exact solution of the Riemann problem | p. 40 |
Methods based on approximate Riemann problem solvers | p. 43 |
Courant-Isaacson-Rees-type methods | p. 43 |
Roe's scheme | p. 55 |
The Osher numerical scheme | p. 58 |
Generalized Riemann problem | p. 65 |
The Godunov method of the second order | p. 67 |
Multidimensional schemes and their stability conditions | p. 72 |
Reconstruction procedures and slope limiters | p. 76 |
Preliminary remarks | p. 76 |
TVD schemes | p. 78 |
Monotone and limiting reconstructions | p. 80 |
Genuine TVD and TVD limiting reconstructions | p. 87 |
TVD limiters of nonsymmetric stencil | p. 92 |
Multidimensional reconstruction | p. 94 |
Boundary conditions for hyperbolic systems | p. 101 |
General notions | p. 101 |
Nonreflecting boundary conditions | p. 102 |
Evolutionary boundary conditions | p. 107 |
Shock-fitting methods | p. 109 |
Floating shock fitting | p. 109 |
Shock fitting on moving grids | p. 113 |
Entropy correction procedures | p. 115 |
Final remarks | p. 119 |
Gas Dynamic Equations | p. 121 |
Systems of governing equations | p. 121 |
Two-temperature gas dynamic equations | p. 124 |
The mixture of ideal gases in chemical nonequilibrium | p. 127 |
The Godunov method for gas dynamic equations | p. 129 |
Exact solution of the Riemann problem | p. 132 |
Elementary solution 1: Shock wave | p. 132 |
Elementary solution 2: Contact discontinuity | p. 136 |
Elementary solution 3: Rarefaction wave | p. 136 |
General exact solution | p. 139 |
An arbitrary EOS | p. 147 |
Approximate Riemann problem solvers | p. 151 |
The Courant-Isaacson-Rees method for an arbitrary EOS | p. 152 |
Computation of shock-induced phenomena by the CIR method | p. 154 |
The CIR-simulation of jet-like structures in laser plasma | p. 158 |
Roe's method | p. 163 |
Roe's Riemann problem solver for an arbitrary EOS | p. 169 |
Osher-Solomon numerical scheme | p. 171 |
Shock-fitting methods | p. 175 |
Discontinuities as boundaries of the computational region | p. 175 |
Floating shock-fitting procedures | p. 186 |
Shock-fitting on moving grids | p. 189 |
Self-adjusting grids | p. 190 |
Stationary gas dynamics | p. 197 |
Systems of governing equations | p. 197 |
The Godunov method. The CIR and Roe's schemes | p. 201 |
Exact solution of the Riemann problem | p. 203 |
General exact solution | p. 212 |
Solar wind - interstellar medium interaction | p. 213 |
Physical formulation of the problem | p. 213 |
Nonreflecting boundary conditions | p. 218 |
Numerical results | p. 221 |
A note on Godunov-type methods for relativistic hydrodynamics | p. 224 |
Shallow Water Equations | p. 225 |
System of governing equations | p. 225 |
The Godunov method for shallow water equations | p. 228 |
Exact solution of the Riemann problem | p. 231 |
Elementary solution 1: Hydraulic jump | p. 231 |
Elementary solution 2: Tangential discontinuity | p. 235 |
Elementary solution 3: Riemann wave | p. 235 |
General exact solution | p. 237 |
Results of numerical analysis | p. 245 |
Approximate Riemann problem solvers | p. 257 |
The CIR method | p. 257 |
Roe's method | p. 258 |
The Osher-Solomon solver | p. 261 |
Stationary shallow water equations | p. 262 |
System of governing equations | p. 263 |
The Godunov method. The CIR and Roe's schemes | p. 265 |
Exact solution of the Riemann problem | p. 266 |
General exact solution | p. 275 |
Magnetohydrodynamic Equations | p. 277 |
MHD system in the conservation-law form | p. 277 |
Classification of MHD discontinuities | p. 285 |
Evolutionary MHD shocks | p. 288 |
Evolutionary diagram | p. 288 |
Convenient relations on MHD shocks | p. 290 |
Evolutionarity of perpendicular, parallel, and singular shocks | p. 291 |
Jouget points | p. 294 |
High-resolution numerical schemes for MHD equations | p. 295 |
The Osher-type method | p. 296 |
Piecewise-parabolic method | p. 297 |
Roe's characteristic decomposition method | p. 298 |
Numerical tests with the Roe-type scheme | p. 305 |
Modified MHD system | p. 323 |
Shock-capturing approach and nonevolutionary solutions in MHD | p. 328 |
Preliminary remarks | p. 328 |
Simplified MHD equations and related discontinuities | p. 332 |
Shock structure in solutions of the simplified system | p. 333 |
Nonstationary processes in the structure of nonevolutionary shock waves | p. 335 |
Numerical experiments based on the full set of MHD equations | p. 337 |
Numerical disintegration of a compound wave | p. 339 |
Strong background magnetic field | p. 345 |
Elimination of numerical magnetic charge | p. 348 |
Preliminary remarks | p. 348 |
Application of the vector potential | p. 349 |
The use of an artificial scalar potential | p. 350 |
Application of the modified MHD system | p. 351 |
Application of staggered grids | p. 352 |
Solar wind interaction with the magnetized interstellar medium | p. 356 |
Statement of the problem | p. 357 |
Numerical algorithm | p. 359 |
Numerical results: axisymmetric case | p. 363 |
Numerical results: rotationally perturbed flow | p. 368 |
A note on the MHD flow over an infinitely conducting cylinder | p. 372 |
Numerical results: three-dimensional modelling | p. 374 |
Solid Dynamics Equations | p. 379 |
System of governing equations | p. 379 |
Solid dynamics with an arbitrary EOS | p. 380 |
Conservative form of elastoviscoplastic solid dynamics | p. 392 |
Dynamics of thin shells | p. 396 |
CIR method for the calculation of solid dynamics problems | p. 399 |
Numerical simulation of spallation phenomena | p. 404 |
CIR method for studying the dynamics of thin shells | p. 410 |
The Klein-Gordon equation | p. 415 |
Dynamics equations of cylindrical shells | p. 416 |
Dynamics equations of orthotropic shells | p. 418 |
Selection of rapidly oscillating components | p. 419 |
Nonclassical Discontinuities and Solutions of Hyperbolic Systems | p. 423 |
Evolutionary conditions in nonclassical cases | p. 425 |
Structure of fronts. Additional boundary conditions on the fronts | p. 427 |
Equations describing the discontinuity structure | p. 429 |
Formulation of the structure problem and additional assumptions | p. 431 |
Behavior of the solution as [xi] [right arrow] [plus or minus infinity] | p. 432 |
Additional relations on discontinuities | p. 434 |
Main result and its discussion | p. 435 |
A remark on deriving additional relations when condition (7.2.7) is not satisfied | p. 436 |
Hugoniot manifold | p. 438 |
Behavior of the Hugoniot curve in the vicinity of Jouget points and nonuniqueness of solutions of self-similar problems | p. 439 |
Nonlinear small-amplitude waves in anisotropic elastic media | p. 447 |
Basic equations | p. 447 |
Quasilongitudinal waves | p. 449 |
Quasitransverse waves | p. 450 |
Riemann waves | p. 451 |
Shock waves | p. 452 |
Self-similar problems and nonuniqueness of solutions | p. 455 |
Waves in viscoelastic media, vanishing viscosity | p. 456 |
Role of the wave anisotropy and passage to the limit g [right arrow] 0 | p. 459 |
Final conclusions | p. 460 |
Electromagnetic shock waves in ferromagnets | p. 461 |
Long-wave approximation. Elastic analogy | p. 461 |
Structure of electromagnetic shock waves | p. 464 |
The set of admissible discontinuities | p. 470 |
Nonuniqueness of solutions | p. 470 |
Shock waves in composite materials | p. 472 |
Basic equations and the discontinuity structure | p. 472 |
Discontinuity structure; admissible discontinuities | p. 474 |
Case h ] 0 | p. 474 |
Case h [ 0 | p. 478 |
Longitudinal nonlinear waves in elastic rods | p. 479 |
Large-scale model | p. 479 |
Model for moderate-scale motions | p. 481 |
Equations describing the discontinuity structure | p. 482 |
Admissible discontinuities | p. 483 |
More precise large-scale model. Nonuniqueness | p. 486 |
Ionization fronts in a magnetic field | p. 487 |
Large-scale model | p. 487 |
Moderate-scale model | p. 488 |
The set of admissible discontinuities | p. 490 |
The simplest self-similar problem | p. 494 |
Variation of the gas velocity across ionization fronts | p. 495 |
Constructing the solution of the piston problem | p. 500 |
Discussion | p. 501 |
Bibliography | p. 503 |
Index | p. 535 |
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