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| Авторизация |
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| Ascher U., Petzold L. — Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations |
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| Предметный указатель |
Absolute stability 44—48
Absolute stability, implicit Runge — Kutta methods 102
Absolute stability, plotting the region of 88 141
Absolute stability, region of 45
Absolute stability, region of, explicit Runge — Kutta methods 87—90
Absolute stability, region of, multistep methods 141—142
Accuracy, order of 40
Adams methods 124—129
Adams methods, 0-stability 140
Adams methods, absolute stability 142
Adams methods, Adams — Bashforth (explicit) method 127
Adams methods, Adams — Moulton (implicit) method 127
Algebraic variables (DAE) 234
Almost block diagonal 207
Artificial diffusion 226
Automatic differentiation 66 289
Autonomous 5 35 82
b-convergence 109
Backward differentiation formulae (BDF), methods 129
Backward differentiation formulae (BDF), methods, 0-stability 140
Backward differentiation formulae (BDF), methods, DAE 266—268
Backward Euler method 49—58 128
Backward Euler method, DAE 263—266
Backward Euler method, region of absolute stability 52
Backward Euler method, solution of nonlinear system 52
Bifurcation diagram 212
Boundary conditions, Dirichlet 225
Boundary conditions, nonseparated 174
Boundary conditions, periodic 163 174 191 222
Boundary conditions, separated 163 203
Boundary conditions, two-point 163
Boundary layer see "Layer"
Boundary value problems (BVPs) 10
Boundary value problems (BVPs) for PDEs 205
Boundary value problems (BVPs), codes, AUTO 222
Boundary value problems (BVPs), codes, COLNEW 221
Boundary value problems (BVPs), codes, COLSYS 221
Boundary value problems (BVPs), codes, MUS 186
Boundary value problems (BVPs), codes, PASVAR 222
Boundary value problems (BVPs), codes, SUPORT 186
Boundary value problems (BVPs), codes, TWPBVP 222
Boundary value problems (BVPs), continuation 210
Boundary value problems (BVPs), damped Newton method 209
Boundary value problems (BVPs), decoupling 186 219
Boundary value problems (BVPs), deferred correction 208
Boundary value problems (BVPs), error estimation 212
Boundary value problems (BVPs), extrapolation 207
Boundary value problems (BVPs), finite difference methods 193—228
Boundary value problems (BVPs), finite difference methods, 0-stability 200
Boundary value problems (BVPs), finite difference methods, collocation 205
Boundary value problems (BVPs), finite difference methods, consistency 200
Boundary value problems (BVPs), finite difference methods, convergence 200
Boundary value problems (BVPs), finite difference methods, solving the linear equations 203
Boundary value problems (BVPs), finite difference methods, stiff problems 214
Boundary value problems (BVPs), infinite interval 187
Boundary value problems (BVPs), mesh selection 212
Boundary value problems (BVPs), midpoint method 194
Boundary value problems (BVPs), multiple shooting method 182—185
Boundary value problems (BVPs), Newton's method 196
Boundary value problems (BVPs), reduced superposition 186
Boundary value problems (BVPs), Riccati method 186
Boundary value problems (BVPs), simple shooting method 177—182
Boundary value problems (BVPs), software 221
Boundary value problems (BVPs), stabilized march method 186
Boundary value problems (BVPs), superposition 186
Boundary value problems (BVPs), trapezoid method 222
Chaos 154
Characteristic polynomial 29
Chemical reaction, BVP example 178 212 223
Collocation methods for BVPs 205
Collocation methods, basic idea 101
Collocation methods, Gauss formulae 100
Collocation methods, Lobatto formulae 100
Collocation methods, order of 102
Collocation methods, order of, for DAEs 270
Collocation methods, projected, for DAE 280
Collocation methods, Radau formulae 100
Collocation methods, relation to implicit Runge — Kutta 101
Compact finite difference methods 225
Compactification see "Multiple shooting method"
Condition number, eigenvalue matrix 48
Condition number, iteration matrix (DAE) 277
Condition number, orthogonal matrix 58
Conservative system 32
Consistency 40
Consistency, BVPs, finite difference methods 200
Consistency, multistep methods 134
Constraint manifold 239
Constraints (DAE), hidden 233
Continuation methods 89 210—212
Continuation methods, arclength 212
Continuous extension 109
Contraction mapping 53
Convection-diffusion equation (PDE) 158
Convergence 40
Convergence of order p 40
Convergence, BDF methods for DAEs 266
Convergence, BVPs, finite difference methods 200
Convergence, calculated rate 79
Convergence, multistep methods 131
Convergence, Runge — Kutta methods 82
Coordinate partitioning (DAE) 253
Corrector formula 144
Crank — Nicolson method for PDEs 69
Damped Newton method 209
Decoupling 172 186 221 264
Decoupling methods (BVP) 172
Decoupling, BVP 219—220
Deferred correction method 208
Degrees of freedom (DAE) 233
Delay differential equation 109 189
Dense output 109
Diagonally implicit Runge — Kutta methods (DIRK) 105
Dichotomy 170 219
Dichotomy, exponential 170
Difference equations 135
Difference operator 40
Differential variables (DAE) 234
Differential-algebraic equations (DAEs) 12 231
Differential-algebraic equations (DAEs), algebraic variables 234
Differential-algebraic equations (DAEs), BDF methods 266
Differential-algebraic equations (DAEs), codes, COLDAE 290
Differential-algebraic equations (DAEs), codes, DASPK 290
Differential-algebraic equations (DAEs), codes, DASSL 290
Differential-algebraic equations (DAEs), codes, MEXX 290
Differential-algebraic equations (DAEs), codes, RADAU 5 290
Differential-algebraic equations (DAEs), consistent initial conditions 233 275
Differential-algebraic equations (DAEs), constraint stabilization 251
Differential-algebraic equations (DAEs), convergence of BDF methods 266
Differential-algebraic equations (DAEs), coordinate partitioning 253
Differential-algebraic equations (DAEs), differential geometric approach 256
Differential-algebraic equations (DAEs), differential variables 234
Differential-algebraic equations (DAEs), direct discretization methods 262
Differential-algebraic equations (DAEs), existence and uniqueness 256
Differential-algebraic equations (DAEs), fully implicit index-1 261
Differential-algebraic equations (DAEs), Hessenberg form 238 256
Differential-algebraic equations (DAEs), Hessenberg index-2 239
Differential-algebraic equations (DAEs), Hessenberg index-3 240
Differential-algebraic equations (DAEs), hidden constraints 233
Differential-algebraic equations (DAEs), higher-index 233
Differential-algebraic equations (DAEs), index reduction and stabilization 247
Differential-algebraic equations (DAEs), index reduction, unstabilized 248
Differential-algebraic equations (DAEs), index, definition of 235
Differential-algebraic equations (DAEs), least squares methods 289
Differential-algebraic equations (DAEs), multistep methods 267
Differential-algebraic equations (DAEs), numerical methods 261
Differential-algebraic equations (DAEs), ODE with constraints 12
Differential-algebraic equations (DAEs), reformulation of higher-index DAEs 247
Differential-algebraic equations (DAEs), regularization 262
Differential-algebraic equations (DAEs), semi-explicit 12 234
| Differential-algebraic equations (DAEs), semi-explicit index-1 238
Differential-algebraic equations (DAEs), simple subsystems 232
Differential-algebraic equations (DAEs), singular 256 262
Differential-algebraic equations (DAEs), stabilization of the constraint 250
Differential-algebraic equations (DAEs), stabilized index-2 formulation 259
Differential-algebraic equations (DAEs), state space formulation 252
Differential-algebraic equations (DAEs), underlying ODE 245 253
Differentiation, automatic 74
Differentiation, symbolic 74
Discontinuity, discretization across 62
Discontinuity, location of 63
Dissipativity 109
Divergence 19 33
Divided differences 125
Drift off the constraint (DAE) 250
Dry friction 64
Dynamical system 17
Dynamical system, discrete 110
Eigenvalue 24
Eigenvector 24
Error estimation, BVPs 212
Error estimation, embedded Runge — Kutta methods 91
Error estimation, global error 94
Error estimation, index-2 DAE 277
Error estimation, multistep methods 149
Error estimation, Runge — Kutta methods 90
Error estimation, step doubling 93
Error, constant (multistep methods) 133
Error, equidistribution 214 221
Error, global 40
Error, local 43
Error, local truncation 40
Error, tolerance absolute and relative 90
Euler method, backward (implicit) 37
Euler method, forward (explicit) 37
Euler method, symplectic 114
Euler method, written as Runge — Kutta 81
Event location 64 109
Explicit, method 39
Explicit, ODE 12
Extraneous roots 136
Extrapolation 108 207
Finite element method 221
Fully implicit index-1 DAEs 261
Functional iteration 52
Functional iteration, multistep methods 143
Fundamental solution 30 166
Fundamental solution in shooting method 178
Fundamental theorem 9
Fundamental theorem, difference methods 41
Gauss collocation 100 103 113 117 119 206 209 212 219 222 290
Gaussian, points 76 102 118
Gaussian, quadrature 76
Global error 40
Global error, estimates of 94 212
Gradient 18 32
Green's function 168 185
Half-explicit Runge — Kutta methods (DAE) 280
Hamiltonian 32
Hamiltonian systems 32 109 114 120
Hamiltonian systems, invariants 249
Hamiltonian systems, preservation of the invariant 254
Hermite interpolation 109
Hessenberg form (DAE) 238 256
Higher-index DAEs 233
Homotopy path 211
Hopf bifurcation 68
Implicit Euler method see "Backward Euler method"
Implicit, method 51
Implicit, ODE 12
Implicit, Runge — Kutta methods 98—108
Implicit, Runge — Kutta methods, implementation of 103 107
Implicit-explicit (IMEX) methods 157
IMSL 66
Incompressible Navier — Stokes equations 239
INDEX 232—246
Index, definition 235
Index, differential 256
Index, perturbation 256
Index, reduction, stabilized index-2 formulation 259
Index, reduction, unstabilized 248
Initial conditions, consistent (DAE) 233
Initial layer see "Layer"
Initial value problem (IVP) 5
Instability of DAE, drift off the constraint 250
Interpolating polynomial and divided differences, review 125
Invariant set 17 35 247 248 252 281
Invariant, integral 118 251
Invariant, ODE with 118
Isolated solution, BVP 165
Isolated solution, IVP 156
Isospectral flow 119
Iteration matrix 54
Jacobian matrix 9 18
Jacobian matrix, difference approximation 54
Kepler problem, modified 120
Kronecker product, review 104
Krylov space methods 153
Lagrange multiplier 12
Lagrange multiplier, DAEs and constrained optimization 239
Layer, boundary 195 206 214 217—219 223
Layer, initial 49 59 61 227
Leapfrog (Verlet) method 114 255
Limit cycle 7 68
Limit set 17
Linearization, local 31
Lipschitz, constant 9
Lipschitz, continuity 9 42
Lobatto collocation 100 103 119 206
Local elimination 207
Local error 43
Local error, control of, in Runge — Kutta methods 90
Local error, estimation by step doubling 93
Local error, relationship to local truncation error 43
Local extrapolation 93
Local truncation error 40 65
Local truncation error, BVPs, finite difference methods 200
Local truncation error, estimation of (multistep methods) 150
Local truncation error, multistep methods 132
Local truncation error, principal term (multistep methods) 150
Local truncation error, relation to local error 65
Long time integration 109
Lyapunov function 35
Mathematica 66
MATLAB 66 98 111
Matrix decompositions, LU 57
Matrix decompositions, QR 58
Matrix decompositions, review 57
Matrix eigenvalues, review 23
Matrix exponential, review 28
Matrix, banded 58
Matrix, sparse 58
Mechanical systems 13 240
Mechanical systems, generalized coordinate partitioning method 256
Mechanical systems, reformulation of higher-index DAEs 247
mesh 37
Mesh function 40
Mesh Reynolds number 226
Mesh selection (BVP) 212
Mesh, locally almost uniform 228
Method of lines 8 14 65 158 211 279
Method of lines, heat equation stability restriction 69
Method of lines, transverse 15
Midpoint method 68 194
Midpoint method, dynamic equivalence to trapezoid method 69
Midpoint method, explicit 75
Midpoint method, explicit, written as Runge — Kutta 81
Midpoint method, staggered 224
Milne's estimate, local truncation error (predictor-corrector methods) 150
Milne's method (multistep method) 139
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