NUMERICAL SOLUTION OF HYPERBOLIC PARTIAL DIFFERENTIAL EQUATIONS
CONTENTS 1 INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS 1 2 SCALAR HYPERBOLIC CONSERVATION LAWS 6 2 1 LINEAR ADVECTION 6 2 1 1 CONSERVATION LAW ON AN UNBOUNDED DOMAIN 6 2 1 2 INTEGRAL FORM OF THE CONSERVATION LAW 8 2 1 3 ADVECTION DIFFUSION EQUATION 9 2 1 4 ADVECTION EQUATION ON A HALF LINE 10 2 1 5 ADVECTION EQUATION ON A FINITE INTERVAL 11 2 2 LINEAR FINITE DIFFERENCE METHODS 12 2 2 1 BASICS OF DISCRETIZATION 12 2 2 2 EXPLICIT UPWIND DIFFERENCES 14 2 2 3 PROGRAMS FOR EXPLICIT UPWIND DIFFERENCES 16 2 2 3 1 FIRST UPWIND DIFFERENCE PROGRAM 16 2 2 3 2 SECOND UPWIND DIFFERENCE PROGRAM 17 2 2 3 3 THIRD UPWIND DIFFERENCE PROGRAM 18 2 2 3 4 FOURTH UPWIND DIFFERENCE PROGRAM 20 2 2 3 5 FIFTH UPWIND DIFFERENCE PROGRAM 21 2 2 4 EXPLICIT DOWNWIND DIFFERENCES 23 2 2 5 IMPLICIT DOWNWIND DIFFERENCES 24 2 2 6 IMPLICIT UPWIND DIFFERENCES 25 2 2 7 EXPLICIT CENTERED DIFFERENCES 26 2 3 MODI ED EQUATION ANALYSIS 30 2 3 1 MODI ED EQUATION ANALYSIS FOR EXPLICIT UPWIND DIFFERENCES 30 2 3 2 MODI ED EQUATION ANALYSIS FOR EXPLICIT DOWNWIND DIFFERENCES 31 2 3 3 MODI ED EQUATION ANALYSIS FOR EXPLICIT CENTERED DIFFERENCES 32 2 3 4 MODI ED EQUATION ANALYSIS LITERATURE 33 2 4 CONSISTENCY STABILITY AND CONVERGENCE 35 2 5 FOURIER ANALYSIS OF FINITE DIFFERENCE SCHEMES 38 2 5 1 CONSTANT COEF CIENT EQUATIONS AND WAVES 39 2 5 2 DIMENSIONLESS GROUPS 40 2 5 3 LINEAR FINITE DIFFERENCES AND ADVECTION 41 2 5 4 FOURIER ANALYSIS OF INDIVIDUAL SCHEMES 44 2 6 L2 STABILITY FOR LINEAR SCHEMES 53 2 7 LAX EQUIVALENCE THEOREM 55 2 8 MEASURING ACCURACY AND EF CIENCY 69 3 NONLINEAR SCALAR LAWS 81 3 1 NONLINEAR HYPERBOLIC CONSERVATION LAWS 81 3 1 1 NONLINEAR EQUATIONS ON UNBOUNDED DOMAINS 81 3 1 2 CHARACTERISTICS 82 3 1 3 DEVELOPMENT OF SINGULARITIES 84 3 1 4 PROPAGATION OF DISCONTINUITIES 85 3 1 5 TRAVELING WAVE PRO LES 89 3 1 6 ENTROPY FUNCTIONS 92 3 1 7 OLEINIK CHORD CONDITION 95 3 1 8 RIEMANN PROBLEMS 97 3 1 9 GALILEAN COORDINATE TRANSFORMATIONS 99 3 2 CASE STUDIES 102 3 2 1 TRAF C FLOW 102 3 2 2 MISCIBLE DISPLACEMENT MODEL 103 3 2 3 BUCKLEY LEVERETT MODEL 105 3 3 FIRST ORDER FINITE DIFFERENCE METHODS 111 3 3 1 EXPLICIT UPWIND DIFFERENCES 111 3 3 2 LAX FRIEDRICHS SCHEME 112 3 3 3 TIMESTEP SELECTION 117 3 3 4 RUSANOV S SCHEME 118 3 3 5 GODUNOV S SCHEME 120 3 3 6 COMPARISON OF LAX FRIEDRICHS GODUNOV AND RUSANOV 124 3 4 NONRE ECTING BOUNDARY CONDITIONS 125 3 5 LAX WENDROFF PROCESS 129 3 6 OTHER SECOND ORDER SCHEMES 132 4 NONLINEAR HYPERBOLIC SYSTEMS 135 4 1 THEORY OF HYPERBOLIC SYSTEMS 135 4 1 1 HYPERBOLICITY AND CHARACTERISTICS 135 4 1 2 LINEAR SYSTEMS 139 4 1 3 FRAMES OF REFERENCE 140 4 1 3 1 USEFUL IDENTITIES 141 4 1 3 2 CHANGE OF FRAME OF REFERENCE FOR CONSERVATION LAWS 143 4 1 3 3 CHANGE OF FRAME OF REFERENCE FOR PROPAGATING DISCONTINUITIES 145 4 1 4 RANKINE HUGONIOT JUMP CONDITION 146 4 1 5 LAX ADMISSIBILITY CONDITIONS 150 4 1 6 ASYMPTOTIC BEHAVIOR OF HUGONIOT LOCI 152 4 1 7 CENTERED RAREFACTIONS 156 4 1 8 RIEMANN PROBLEMS 159 4 1 9 RIEMANN PROBLEM FOR LINEAR SYSTEMS 159 4 1 10 RIEMANN PROBLEM FOR SHALLOW WATER 162 4 1 11 ENTROPY FUNCTIONS 164 4 2 UPWIND SCHEMES 176 4 2 1 LAX FRIEDRICHS SCHEME 176 4 2 2 RUSANOV SCHEME 179 4 2 3 GODUNOV SCHEME 179 4 3 CASE STUDY MAXWELL S EQUATIONS 183 4 3 1 CONSERVATION LAWS 183 4 3 2 CHARACTERISTIC ANALYSIS 184 4 4 CASE STUDY GAS DYNAMICS 186 4 4 1 CONSERVATION LAWS 187 4 4 2 THERMODYNAMICS 187 4 4 3 CHARACTERISTIC ANALYSIS 188 4 4 4 ENTROPY FUNCTION 190 4 4 5 CENTERED RAREFACTION CURVES 192 4 4 6 JUMP CONDITIONS 194 4 4 7 RIEMANN PROBLEM 200 4 4 8 RE ECTING WALLS 205 4 5 CASE STUDY MAGNETOHYDRODYNAMICS MHD 208 4 5 1 CONSERVATION LAWS 208 4 5 2 CHARACTERISTIC ANALYSIS 209 4 5 3 ENTROPY FUNCTION 218 4 5 4 CENTERED RAREFACTION CURVES 218 4 5 5 JUMP CONDITIONS 220 4 6 CASE STUDY FINITE DEFORMATION IN ELASTIC SOLIDS 221 4 6 1 EULERIAN FORMULATION OF EQUATIONS OF MOTION FOR SOLIDS 221 4 6 2 LAGRANGIAN FORMULATION OF EQUATIONS OF MOTION FOR SOLIDS 222 4 6 3 CONSTITUTIVE LAWS 223 4 6 4 CONSERVATION FORM OF THE EQUATIONS OF MOTION FOR SOLIDS 225 4 6 5 JUMP CONDITIONS FOR ISOTHERMAL SOLIDS 226 4 6 6 CHARACTERISTIC ANALYSIS FOR SOLIDS 227 4 7 CASE STUDY LINEAR ELASTICITY 233 4 8 CASE STUDY VIBRATING STRING 235 4 8 1 CONSERVATION LAWS 235 4 8 2 CHARACTERISTIC ANALYSIS 237 4 8 3 JUMP CONDITIONS 238 4 8 4 LAX ADMISSIBILITY CONDITIONS 240 4 8 5 ENTROPY FUNCTION 240 4 8 6 WAVE FAMILIES FOR CONCAVE TENSION 241 4 8 7 WAVE FAMILY INTERSECTIONS 245 4 8 8 RIEMANN PROBLEM SOLUTION 249 4 9 CASE STUDY PLASTICITY 255 4 9 1 LAGRANGIAN EQUATIONS OF MOTION 255 4 9 2 CONSTITUTIVE LAWS 256 4 9 3 CENTERED RAREFACTIONS 258 4 9 4 HUGONIOT LOCI 259 4 9 5 ENTROPY FUNCTION 261 4 9 6 RIEMANN PROBLEM 261 4 10 CASE STUDY POLYMER MODEL 267 4 10 1 CONSTITUTIVE LAWS 268 4 10 2 CHARACTERISTIC ANALYSIS 269 4 10 3 JUMP CONDITIONS 270 4 10 4 RIEMANN PROBLEM SOLUTION 271 4 11 CASE STUDY THREE PHASE BUCKLEY LEVERETT FLOW 274 4 11 1 CONSTITUTIVE MODELS 274 4 11 2 CHARACTERISTIC ANALYSIS 276 4 11 3 UMBILIC POINT 277 4 11 4 ELLIPTIC REGIONS 277 4 12 CASE STUDY SCHAEFFER SCHECHTER SHEARER SYSTEM 278 4 13 APPROXIMATE RIEMANN SOLVERS 284 4 13 1 DESIGN OF APPROXIMATE RIEMANN SOLVERS 284 4 13 2 ARTI CIAL DIFFUSION 291 4 13 3 RUSANOV SOLVER 293 4 13 4 WEAK WAVE RIEMANN SOLVER 294 4 13 5 COLELLA GLAZ RIEMANN SOLVER 296 4 13 6 OSHER SOLOMON RIEMANN SOLVER 298 4 13 7 BELL COLELLA TRANGENSTEIN APPROXIMATE RIEMANN PROBLEM SOLVER 299 4 13 8 ROE RIEMANN SOLVER 304 4 13 9 HARTEN HYMAN MODI CATION OF THE ROE SOLVER 313 4 13 10HARTEN LAX VAN LEER SCHEME 315 4 13 11HLL SOLVERS WITH TWO INTERMEDIATE STATES 317 4 13 12APPROXIMATE RIEMANN SOLVER RECOMMENDATIONS 320 5 METHODS FOR SCALAR LAWS 326 5 1 CONVERGENCE 326 5 1 1 CONSISTENCY AND ORDER 326 5 1 2 LINEAR METHODS AND STABILITY 328 5 1 3 CONVERGENCE OF LINEAR METHODS 330 5 2 ENTROPY CONDITIONS AND DIFFERENCE APPROXIMATIONS 331 5 2 1 BOUNDED CONVERGENCE 331 5 2 2 MONOTONE SCHEMES 341 5 3 NONLINEAR STABILITY 353 5 3 1 TOTAL VARIATION 353 5 3 2 TOTAL VARIATION STABILITY 354 5 3 3 OTHER STABILITY NOTIONS 357 5 4 PROPAGATION OF NUMERICAL DISCONTINUITIES 359 5 5 MONOTONIC SCHEMES 361 5 5 1 SMOOTHNESS MONITOR 361 5 5 2 MONOTONIZATIONS 362 5 5 3 MUSCL SCHEME 364 5 6 DISCRETE ENTROPY CONDITIONS 367 5 7 E SCHEMES 368 5 8 TOTAL VARIATION DIMINISHING SCHEMES 370 5 8 1 SUF CIENT CONDITIONS FOR DIMINISHING TOTAL VARIATION 370 5 8 2 HIGHER ORDER TVD SCHEMES FOR LINEAR ADVECTION 375 5 8 3 EXTENSION TO NONLINEAR SCALAR CONSERVATION LAWS 379 5 9 SLOPE LIMITER SCHEMES 383 5 9 1 EXACT INTEGRATION FOR CONSTANT VELOCITY 384 5 9 2 PIECEWISE LINEAR RECONSTRUCTION 386 5 9 3 TEMPORAL QUADRATURE FOR FLUX INTEGRALS 388 5 9 4 CHARACTERISTIC TRACING 389 5 9 5 FLUX EVALUATION 390 5 9 6 NON RE ECTING BOUNDARIES WITH THE MUSCL SCHEME 391