MATHEMATICAL ASPECTS OF NUMERICAL SOLUTION OF HYPERBOLIC SYSTEMS

Transcription

1 K CHAPMAN & HALL/CRC Monographs and Surveys in Pure and Applied Mathematics I 18 MATHEMATICAL ASPECTS OF NUMERICAL SOLUTION OF HYPERBOLIC SYSTEMS ANDREI G. KULIKOVSKII NIKOLAI V. POGORELOV ANDREI YU. SEMENOV CHAPMAN & HALL/CRC Boca Raton London New York Washington, D.C.

2 Contents 1 Hyperbolic Systems of Partial Differential Equations Quasilinear systems Hyperbolic systems of quasilinear differential equations Definitions Systems of conservation laws Mechanical examples Nonstationary equations of gas dynamics Stationary Euler equations Shallow water equations Equations of ideal magnetohydrodynamics Elasticity equations Properties of solutions Classical solutions Generalized solutions Small-amplitude shocks Evolutionary conditions for shocks Entropy behavior on discontinuities Disintegration of a small arbitrary discontinuity 31 2 Numerical Solution of Quasilinear Hyperbolic Systems "introduction Methods based on the exact solution of the Riemann problem The Godunov method of the first order Exact solution of the Riemann problem Methods based on approximate Riemann problem solvers Courant-Isaacson-Rees-type methods Roe's scheme The Osher numerical scheme Generalized Riemann problem The Godunov method of the second order Multidimensional scheihes and their stability conditions Reconstruction procedures and slope limiters Preliminary remarks TVD schemes Monotone and limiting reconstructions 80

3 2.7.4 Genuine TVD and TVD limiting reconstructions TVD limiters of nonsymmetric stencil Multidimensional reconstruction Boundary conditions for hyperbolic systems General notions Nonreflecting boundary conditions Evolutionary boundary conditions Shock-fitting methods Floating shock fitting Shock fitting on moving grids Entropy correction procedures Final remarks 119 Gas Dynamic Equations Systems of governing equations Two-temperature gas dynamic equations The mixture of ideal gases in chemical nonequilibrium The Godunov method for gas dynamic equations Exact solution of the Riemann problem Elementary solution 1: Shockwave Elementary solution 2: Contact discontinuity Elementary solution 3: Rarefaction wave General exact solution An arbitrary EOS Approximate Riemann problem solvers The Courant-Isaacson-Rees method for an arbitrary EOS Computation of shock-induced phenomena by the CIR method The CIR-simulation of jet-like structures in laser plasma Roe's method Roe's Riemann problem solver for an arbitrary EOS Osher-Solomon numerical scheme Shock-fitting methods Discontinuities as boundaries of the computational region Floating shock-fitting procedures Shock-fitting on moving grids Self-adjusting grids Stationary gas dynamics Systems of governing equations The Godunov method. The CIR and Roe's schemes Exact solution of the Riemann problem General exact solution Solar wind - interstellar medium interaction Physical formulation of the problem Nonreflecting boundary conditions Numerical results 221

4 3.7.4 A note on Godunov-type methods for relativistic hydrodynamics Shallow Water Equations System of governing equations The Godunov method for shallow water equations Exact solution of the Riemann problem v Elementary solution 1: Hydraulic jump Elementary solution 2: Tangential discontinuity Elementary solution 3: Riemann wave General exact solution Results of numerical analysis Approximate Riemann problem solvers The CIR method Roe's method The Osher-Solomon solver Stationary shallow water equations System of governing equations The Godunov method. The CIR and Roe's schemes Exact solution of the Riemann problem General exact solution 275 Magnetohydrodynamic Equations MHD system in the conservation-law form Classification of MHD discontinuities Evolutionary MHD shocks Evolutionary diagram Convenient relations on MHD shocks Evolutionarity of perpendicular, parallel, and singular shocks Jouget points High-resolution numerical schemes for MHD equations The Osher-type method Piecewise-parabolic method Roe's characteristic decomposition method Numerical tests with the Roe-type scheme Modified MHD system Shock-capturing approach and nonevolutionary solutions in MHD Preliminary remarks Simplified MHD equations and related discontinuities Shock structure in solutions of the simplified system Nonstationary processes in the structure of nonevolutionary shock waves Numerical experiments based on the full set of MHD equations Numerical disintegration of a compound wave Strong background magnetic field Elimination of numerical magnetic charge 348

5 Xll Preliminary remarks Application of the vector potential The use of an artificial scalar potential _ Application of the modified MHD system Application of staggered grids Solar wind interaction with the magnetized interstellar medium Statement of the problem Numerical algorithm Numerical results: axisymmetric case Numerical results: rotationally perturbed flow A note on the MHDflowover an infinitely conducting cylinder Numerical results: three-dimensional modelling Solid Dynamics Equations System of governing equations Solid dynamics with an arbitrary EOS Conservative form of elastoviscoplastic solid dynamics Dynamics of thin shells CIR method for the calculation of solid dynamics problems Numerical simulation of spallation phenomena CIR method for studying the dynamics of thin shells The Klein-Gordon equation Dynamics equations of cylindrical shells.... x Dynamics equations of orthotropic shells Selection of rapidly oscillating components Nonclassical Discontinuities and Solutions of Hyperbolic Systems Evolutionary conditions in nonclassical cases Structure of fronts. Additional boundary conditions on the fronts Equations describing the discontinuity structure Formulation of the structure problem and additional assumptions :2.3 Behavior of the solution as -+ ±oo Additional relations on discontinuities Main result and its discussion A remark on deriving additional relations when condition (7.2.7) is not satisfied ' Hugoniot manifold Behavior of the Hugoniot curve in the vicinity of Jouget points and nonuniqueness of solutions of self-similar problems Nonlinear small-amplitude waves in anisotropic elastic media Basic equations Quasilongitudinal waves Quasitransverse waves Riemann waves Shockwaves 452

6 7.4.6 Self-similar problems and nonuniqueness of solutions Waves in viscoelastic media, vanishing viscosity Role of the wave anisotropy and passage to the limit g -> Final conclusions Electromagnetic shock waves in ferromagnets Long-wave approximation. Elastic analogy Structure of electromagnetic shock waves The set of admissible discontinuities Nonuniqueness of solutions Shock waves in composite materials Basic equations and the discontinuity structure Discontinuity structure; admissible discontinuities Casefc> Casefc< Longitudinal nonlinear waves in elastic rods Large-scale model Model for moderate-scale motions Equations describing the discontinuity structure Admissible discontinuities More precise large-scale model. Nonuniqueness Ionization fronts in a magnetic field Large-scale model Moderate-scale model The set of admissible discontinuities The simplest self-similar problem Variation of the gas velocity across ionization fronts Constructing the solution of the piston problem Discussion 501 Bibliography 503 Index 535 xm