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| Petzold L. — Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations |
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| Предметный указатель |
Absolute stability 42—47
Absolute stability, implicit Runge — Kutta methods 104
Absolute stability, plotting the region of 89 144
Absolute stability, region of 43
Absolute stability, region of, explicit Runge — Kutta methods 89—91
Absolute stability, region of, multistep methods 143—145
Accuracy, order of 38
Adams methods 128—131
Adams methods, 0-stability 143
Adams methods, absolute stability 144
Adams methods, Adams — Bashforth (explicit) method 129
Adams methods, Adams — Moulton (implicit) method 129
Algebraic variables (DAE) 234
Almost block diagonal 208
Artificial diffusion 228
Automatic differentiation 65 292
Autonomous 3 33 83
b-convergence 110
Backward differentiation formulae see "BDF"
Backward Euler method 47—56 130
Backward Euler method, DAE 265—268
Backward Euler method, region of absolute stability 50
Backward Euler method, solution of nonlinear system 50
BDF methods 131—132
BDF methods, 0-stability 143
BDF methods, DAE 268—270
Bifurcation diagram 212
Boundary conditions, Dirichlet 227
Boundary conditions, non-separate 174
Boundary conditions, periodic 163 174 192 223
Boundary conditions, separated 163 204
Boundary conditions, two-point 163
Boundary layer see "Layer"
Boundary value problems (BVP) 8
Boundary value problems (BVP) for PDEs 206
Boundary value problems (BVP), continuation 211
Boundary value problems (BVP), damped Newton method 210
Boundary value problems (BVP), decoupling 186 220
Boundary value problems (BVP), deferred correction 208
Boundary value problems (BVP), error estimation 213
Boundary value problems (BVP), extrapolation 208
Boundary value problems (BVP), finite difference methods 193—230
Boundary value problems (BVP), finite difference methods, 0-stability 201
Boundary value problems (BVP), finite difference methods, collocation 206
Boundary value problems (BVP), finite difference methods, consistency 201
Boundary value problems (BVP), finite difference methods, convergence 201
Boundary value problems (BVP), finite difference methods, solving the linear equations 204
Boundary value problems (BVP), finite difference methods, stiff problems 215
Boundary value problems (BVP), infinite interval 188
Boundary value problems (BVP), mesh selection 213
Boundary value problems (BVP), midpoint method 194
Boundary value problems (BVP), multiple shooting method 183—186
Boundary value problems (BVP), Newton's method 197
Boundary value problems (BVP), reduced superposition 186
Boundary value problems (BVP), Riccati method 186
Boundary value problems (BVP), simple shooting method 177—182
Boundary value problems (BVP), software 223
Boundary value problems (BVP), stabilized march method 186
Boundary value problems (BVP), superposition 186
Boundary value problems (BVP), trapezoid method 223
BVP codes, AUTO 223
BVP codes, COLNEW 223
BVP codes, COLSYS 223
BVP codes, MUS 187
BVP codes, PASVAR 223
BVP codes, SUPORT 187
BVP codes, TWPBVP 223
Chaos 158
Characteristic polynomial 26
Chemical reaction, BVP example 178 213 224
Collocation methods for BVPs 206
Collocation methods, basic idea 102
Collocation methods, Gauss formulae 101
Collocation methods, Lobatto formulae 102
Collocation methods, order for DAEs 272
Collocation methods, order of 103
Collocation methods, projected, for DAE 281
Collocation methods, Radau formulae 101
Collocation methods, relation to implicit Runge — Kutta 102
Compact finite difference methods 226
Compactification, multiple shooting method 191
Condition number, eigenvalue matrix 46
Condition number, iteration matrix (DAE) 278
Condition number, orthogonal matrix 56
Conservative system 29
Consistency 38
Consistency, BVPs, finite difference methods 201
Consistency, multistep methods 137
Constraint manifold 240
Constraints (DAE), hidden 234
Continuation methods 90 211—213
Continuation methods, arclength 213
Continuous extension 110
Contraction mapping 51
Convection-diffusion equation (PDE) 161
Convergence 38
Convergence of order p 39
Convergence, BDF methods for DAEs 268
Convergence, BVPs, finite difference methods 201
Convergence, calculated rate 79
Convergence, multistep methods 134
Convergence, Runge — Kutta methods 83
Coordinate partitioning (DAE) 254
Corrector formula 147
Crank — Nicolson method for PDEs 69
DAE codes, COLDAE 293
DAE codes, DASPK 293
DAE codes, DASSL 293
DAE codes, MEXX 293
DAE codes, RADAU5 293
Damped Newton method 210
Decoupling 172 186 222 267
Decoupling methods (BVP) 172
Decoupling, BVP 220—221
Deferred correction method 208
Degrees of freedom (DAE) 233
Delay differential equation 111 189
Dense output 110
Diagonally implicit Runge — Kutta methods (DIRK) 106
Dichotomy 170 220
Dichotomy, exponential 170
Difference equations 137
Difference operator 38
Differential variables (DAE) 234
Differential-algebraic equations (DAE) 10 231
Differential-algebraic equations (DAE), algebraic variables 234
Differential-algebraic equations (DAE), BDF methods 268
Differential-algebraic equations (DAE), consistent initial conditions 233 276
Differential-algebraic equations (DAE), constraint stabilization 253
Differential-algebraic equations (DAE), convergence of BDF methods 268
Differential-algebraic equations (DAE), coordinate partitioning 254
Differential-algebraic equations (DAE), differential geometric approach 256
Differential-algebraic equations (DAE), differential variables 234
Differential-algebraic equations (DAE), direct discretization methods 264
Differential-algebraic equations (DAE), existence and uniqueness 256
Differential-algebraic equations (DAE), fully-implicit index-1 263
Differential-algebraic equations (DAE), Hessenberg form 238 257
Differential-algebraic equations (DAE), Hessenberg index-2 239
Differential-algebraic equations (DAE), Hessenberg index-3 240
Differential-algebraic equations (DAE), hidden constraints 234
Differential-algebraic equations (DAE), higher-index 234
Differential-algebraic equations (DAE), index reduction and stabilization 247
Differential-algebraic equations (DAE), index reduction, unstabilized 249
Differential-algebraic equations (DAE), index, definition 235
Differential-algebraic equations (DAE), least squares methods 291
Differential-algebraic equations (DAE), multistep methods 269
Differential-algebraic equations (DAE), numerical methods 263
Differential-algebraic equations (DAE), ODE with constraints 10
Differential-algebraic equations (DAE), reformulation of higher-index DAEs 248
Differential-algebraic equations (DAE), regularization 264
| Differential-algebraic equations (DAE), semi-explicit 10 234
Differential-algebraic equations (DAE), semi-explicit index-1 238
Differential-algebraic equations (DAE), simple subsystems 232
Differential-algebraic equations (DAE), singular 257 264
Differential-algebraic equations (DAE), stabilization of the constraint 251
Differential-algebraic equations (DAE), stabilized index-2 formulation 260
Differential-algebraic equations (DAE), state space formulation 253
Differential-algebraic equations (DAE), underlying ODE 246 254
Differentiation, automatic 75
Differentiation, symbolic 75
Discontinuity, discretization across 61
Discontinuity, location of 63
Dissipativity 111
Divergence 17 30
Divided differences 126
Drift off the constraint (DAE) 251
Dry friction 63
Dynamical system 15
Dynamical system, discrete 112
Eigenvalue 20
Eigenvector 20
Error estimation, BVPs 213
Error estimation, embedded Runge — Kutta methods 92
Error estimation, global error 95
Error estimation, index-2 DAE 279
Error estimation, multistep methods 152
Error estimation, Runge — Kutta methods 91
Error estimation, step doubling 94
Error, constant (multistep methods) 136
Error, equidistribution 215 222
Error, global 39
Error, local 41
Error, local truncation 38
Error, tolerance, absolute and relative 91
Euler method, backward (implicit) 35
Euler method, forward (explicit) 35
Euler method, symplectic 116
Euler method, written as Runge — Kutta 81
Event location 63 111
Explicit method 37
Explicit ODE 9
Extraneous roots 139
Extrapolation 110 208
Finite element method 222
Fully-implicit index-1 DAEs 263
Functional iteration 50
Functional iteration, multistep methods 146
Fundamental solution 26 166
Fundamental solution in shooting method 178
Fundamental theorem 6
Fundamental theorem, difference methods 39
Gauss collocation 101 104 115 119 121 207 210 213 220 223 293
Gaussian points 76 104 120
Gaussian quadrature 76
Global error 39
Global error, estimates of 95 213
Gradient 17 29
Green's function 168 185
Half-explicit Runge — Kutta methods (DAE) 281
Hamiltonian 29
Hamiltonian systems 29 111 116 123
Hamiltonian systems, invariants 250
Hamiltonian systems, preservation of the invariant 255
Hermite interpolation 111
Hessenberg form (DAE) 238 257
Higher index DAEs 234
Homotopy path 211
Hopf bifurcation 67
Implicit Euler method see "Backward Euler method"
Implicit method 49
Implicit ODE 10
Implicit Runge — Kutta methods 101—109
Implicit Runge — Kutta methods, implementation 105 109
Implicit-explicit (IMEX) methods 161
IMSL 66
Incompressible Navier — Stokes equations 239
INDEX 232—247
Index, definition 235
Index, differential 257
Index, perturbation 257
Index, reduction, stabilized index-2 formulation 260
Index, reduction, unstabilized 249
Initial conditions, consistent (DAE) 233
Initial layer see "Layer"
Initial value problem (IVP) 3
Instability, DAE, drift off the constraint 251
Interpolating polynomial and divided differences, review 126
Invariant set 15 32 247 249 253 282
Invariant, integral 120 251
Invariant, ODE with 120
Isolated solution (BVP) 165
Isolated solution (IVP) 159
Isospectral flow 121
Iteration matrix 53
Jacobian matrix 7 17
Jacobian matrix, difference approximation 54
Kepler problem, modified 123
Kronecker product, review 105
Krylov space methods 156
Lagrange multiplier 10
Lagrange multiplier, DAEs and constrained optimization 240
Layer, boundary 195 207 215 218—220 224
Layer, initial 47 58 60 229
Leapfrog (Verlet) method 116 256
Limit cycle 5 67
Limit set 15
Linearization, local 28
Lipschitz constant 7
Lipschitz continuity 6 40
Lobatto collocation 102 104 121 207
Local elimination 208
Local error 41
Local error, control of, in Runge — Kutta methods 91
Local error, estimation by step doubling 94
Local error, relationship to local truncation error 41
Local extrapolation 94
Local truncation error 38 64
Local truncation error, BVPs, finite difference methods 201
Local truncation error, estimation of (multistep methods) 153
Local truncation error, multistep methods 134
Local truncation error, principal term (multistep methods) 154
Local truncation error, relation to local error 64
Long time integration 111
Lyapunov function 32
Mathematica 65
MATLAB 65 99 113
Matrix decompositions, LU 55
Matrix decompositions, QR 56
Matrix decompositions, review 54
Matrix eigenvalues, review 19
Matrix exponential, review 24
Matrix, banded 56
Matrix, sparse 56
Mechanical systems 11 240
Mechanical systems, generalized coordinate partitioning method 257
Mechanical systems, reformulation of higher-index DAEs 248
mesh 35
Mesh function 38
Mesh Reynolds number 227
Mesh selection (BVP) 213
Mesh, locally almost uniform 229
Method of lines 6 12 64 161 212 280
Method of lines, heat equation stability restriction 69
Method of lines, transverse 13
Midpoint method 68 194
Midpoint method, dynamic equivalence to trapezoid method 68
Midpoint method, explicit 78
Midpoint method, explicit, written as Runge — Kutta 82
Midpoint method, staggered 225
Milne's estimate, local truncation error (predictor-corrector methods) 153
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