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

Автор: 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: 1997

Количество страниц: 315

Добавлена в каталог: 13.12.2013

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