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Ascher U., Petzold L. — Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Ascher U., Petzold L. — Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations

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Название: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations

Авторы: Ascher U., Petzold L.

Аннотация:

Designed for those people who want to gain a practical knowledge of modern techniques, this book contains all the material necessary for a course on the numerical solution of differential equations. Written by two of the field's leading authorities, it provides a unified presentation of initial value and boundary value problems in ODEs as well as differential-algebraic equations. The approach is aimed at a thorough understanding of the issues and methods for practical computation while avoiding an extensive theorem-proof type of exposition. It also addresses reasons why existing software succeeds or fails. This is a practical and mathematically well informed introduction that emphasizes basic methods and theory, issues in the use and development of mathematical software, and examples from scientific engineering applications. Topics requiring an extensive amount of mathematical development are introduced, motivated, and included in the exercises, but a complete and rigorous mathematical presentation is referenced rather than included.


Язык: en

Рубрика: Computer science/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1998

Количество страниц: 332

Добавлена в каталог: 12.02.2014

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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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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