.CONTENTS
.Introduction.to.the.Series..............................................xi
.Preface...............................................................xiii
.1.Differential.Equations.................................................1
...1.1.From.Continuous.to.Discrete........................................1
...1.2.Stability.Concepts.................................................3
...1.3.Linearization......................................................4
...1.4.Total.Stability....................................................8
...1.5.Hopf.Bifurcation...................................................9
...1.6.Summary.and.Paradigms.............................................12
.Notes...................................................................14
.2.Linear.Difference.Equations.with.Constant.............................15
...2.1.Preliminaries.and.Notations.......................................15
...2.2.The.Case.of.Siinple.Roots.........................................16
...2.3.The.Case.of.Multiple.Roots........................................23
...2.4.The.Nonhomogeneous.Case...........................................27
.......2.4.1.Difference.Equations.in.Matrix.Form.........................30
...2.5.Stability.of.Solutions............................................31
...2.6.Following.a.Particular.Solution...................................36
.......2.6.1.Proof.of.Theorem.2.6.1......................................39
...2.7.Systems.of.Linear.Difference.Equations............................42
.......2.7.1.Linear.Systems.with.Constant.Matrix.........................43
.......2.7.2.The.Ceneral.Linear.Case.....................................44
.......2.7.3.Difference.Equations.with.Matrix.Coefficients...............45
.Notes...................................................................49
.3.Polynomials.and.Toeplitz.Matrices.....................................51
...3.1.Location.of.Zeros.................................................51
.......3.1.1.Conditions.Characterizing.the.Types.of.Polynomials..........53
...3.2.Toeplitz.Band.Matrices.(T-matrices)...............................56
...3.3.Infinite.T-matrices...............................................56
.......3.3.1.Inverse.of.Infinite.T-matrices..............................57
.......3.3.2.Boundary.Locus..............................................61
...3.4.Finite.T-matrices.................................................64
.......3.4.1.Spectrum.of.a.Family.of.Finite.T-matrices...................64
.......3.4.2.Componentwise.Bounds.for.the.Inverses.of.Finite.T-matrices..72
...3.5.Summary...........................................................76
.Notes...................................................................77
.4.Numerical.Methods.for.Initial.Value.Problems..........................79
...4.1.Preliminaries.....................................................79
...4.2.Linear.Multistep.Formulae.(LMF)...................................82
...4.3.LMF.in.Matrix.Form................................................85
...4.4.Convergence.......................................................87
.......4.4.1.Convergence.of.Initial.Value.Methods........................88
.......4.4.2.Convergence.oùBoundary.Value.Methods........................93
...4.5.0(k1,k2)-stability................................................97
...4.6.Fixed-h.Stability.for.Initial.Value.Methods.......................97
...4.7.Fixed-h.Stability.for.Boundary.Value.Methods.....................100
.......4.7.1.Boundary.Locus.and.Related.Questions.......................102
...4.8.Ak1k2-stability.Versus.0(k1,k2)-stability........................105
...4.9.Correct.Use.of.a.Method..........................................107
.......4.9.1.Conditioning.of.T-matrices.and.BVMs........................111
...4.10.Stiff.Problems..................................................113
...4.11.Relative.Stability.and.Unstable.Problems........................114
........4.11.1.Existence.of.Solutions...................................120
.Notes..................................................................120
.5.Generalized.Backward.Differentiation.Formulae........................121
...5.1.BDF.and.Generalized.BDF..........................................121
...5.2.Derivation.of.GBDF...............................................125
.......5.2.1.The.Case.of.a.Nonuniform.Mesh..............................127
.......5.2.2.Solving.Vandermonde.Systems................................128
...5.3.The.Additional.Conditions........................................128
.......5.3.1.Stability.of.the.Discrete.Problem..........................135
...5.4.The.Integration.of.Systems.of.Equations..........................135
.......5.4.1.Stability.Analysis.for.Systems.of.Equations................137
.......5.4.2.The.Behavior.on.the.Imaginary.Axis.........................139
.Notes..................................................................140
.6.Generalized.Adams.Methods............................................143
...6.1.Adams-Moulton.Methods............................................143
.......6.1.1.Derivation.of.the.Adams-Moulton.Formulae...................144
...6.2.Reverse.Adams.Methods............................................146
...6.3.Generalized.Adams.Methods.(GAMs).................................148
.......6.3.1.The.Case.of.a.Nonuniform.Mesh..............................150
...6.4.The.Additional.Conditions........................................152
.......6.4.1.The.Behavior.on.the.Imaginary.Axis.........................154
.Notes..................................................................157
.7.Symmetric.Schemes....................................................159
...7.1.GeneraI.Properties.of.Symmetric.Schemes..........................159
...7.2.Extended.Trapezoidal.Rules.(ETRs)................................162
...7.3.Extended.Trapezoidal.Rules.of.Second.Kind.(ETR2s)................164
.......7.3.1.The.Case.of.a.Nonuniform.Mesh..............................168
.......7.3.2.The.Additional.Conditions..................................168
.......7.3.3.Unsymmetric.ETR2s..........................................170
...7.4.Top.Order.Methods.(TOMs).........................................171
.......7.4.1.The.Additional.Conditions..................................174
.......7.4.2.Variable.Stepsize..........................................175
.......7.4.3.Solving.Confiuent.Vandermonde.Systems......................176
...7.5.Numerical.Examples...............................................177
.......7.5.1.Relative.Stability.Regions.of.Symmetric.Schemes............178
.Notes..................................................................183
.8.Hamiltonian.Problems.................................................185
...8.1.Introduction.....................................................185
...8.2.Symplectic.Methods...............................................188
...8.3.Discrete.Problems................................................194
...8.4.Discrete.Variational.Principle...................................198
...8.5.Time.Reversal.Symmetry.and.Additional.Methods....................201
.......8.5.1.Proof.of.Lemma.8.5.1.......................................206
...8.6.Discrete.Maps....................................................208
...8.7.Numerical.Methods................................................210
.Notes..................................................................212
.9.Boundary.Value.Problems..............................................213
...9.1.Introduction.....................................................213
...9.2.Sensitivity.Analysis.and.Classification.of.Problems..............216
...9.3.Time.Reversal.Symmetry...........................................217
...9.4.Conditioning.of.Linear.Problems..................................222
.......9.4.1.Discrete.BVPs..............................................226
...9.5.Numerical.Methods................................................226
.......9.5.1.The.Contribution.of.Spurious.Roots.........................229
...9.6.Approximating.Continuous.BVPs.by.Means.of.BVMs...................232
.......9.6.1.Numerical.Examples.........................................233
.Notes..................................................................236
.10.Mesh.Selection.Strategies...........................................237
....10.1.Classification.of.Continuous.Problems.and.Stiffness............237
.........10.1.1.The.Scalar.Case.........................................237
.........10.1.2.Systems.of.Equations....................................239
.........10.1.3.Ill.Conditioned.Problems................................242
.........10.1.4.Nonhomogeneous.Problems.................................245
....10.2.Discrete.Problems..............................................247
....10.3.Mesh.Selection.................................................248
.........10.3.1.Control.of.the.Parameters.k_d.and.g_d...................253
.........10.3.2.Estimate.of.the.Precision.Set...........................254
....10.4.Minimization.of.the.Global.Error...............................256
.........10.4.1.Monitoring.the.Truncation.Errors........................260
....10.5.Stability.and.Equidistribution.................................261
....10.6.The.Nonhomogeneous.Case........................................262
....10.7.The.IVP.Case...................................................264
....10.8.Numerical.Examples.............................................271
.Notes..................................................................277
.11.Block.BVMs..........................................................279
....11.1.Introduction...................................................279
....11.2.Matrix.Form....................................................280
....11.3.Block.Version.of.BVMs..........................................282
....11.4.Choosing.the.Additional.Methods................................283
....11.5.B2VMs.and.Runge-Kutta.Schemes..................................286
.........11.5.1.B2VMs.Versus.RK.Schemes.................................288
.........11.5.2.Choosing.the.Blocksize.of.a.B2VM........................289
.........11.5.3.Stability.Properties.of.B2VMs...........................292
....11.6.Block.BVMs.and.Generai.Linear.Methods..........................295
.........11.6.1.Stability.Properties.of.B2VM2s..........................297
.Notes..................................................................299
.12.Parallel.Impiementation.of.B2VMs....................................301
....12.1.Introduction...................................................301
....12.2.The.Paraliel.Algorithm.........................................302
.........12.2.1.Supplementary.Considerations............................305
....12.3.Parallel.Solution.of.Two-poirit.BVPs...........................306
....12.4.Expected.Speed-up..............................................311
.........12.4.1.The.IVP.Case............................................311
.........12.4.2.The.BVP.Case............................................312
....12.5.Parallel.Soiution.of.the.Reduced.System........................313
.........12.5.1.The.IVP.Case............................................314
.........12.5.2.The.BVP.Case............................................317
.........12.5.3.Numerical.Examples......................................320
.Notes..................................................................322
.13.Extensions.and.Applications.to.Speciai.Problems.....................325
....13.1.The.Method.of.Lines............................................325
.........13.1.1.Some.Examples...........................................326
....13.2.Differential.Algebraic.Equations...............................332
.........13.2.1.Numerical.Examples......................................336
....13.3.Delay.Differential.Equations...................................337
.........13.3.1.Numerical.Examples......................................340
....13.4.Multiderivative.BVMs...........................................343
....13.5.Nonlinear.Problems.............................................344
.Notes..................................................................348
.A.Matrices.............................................................349
...A.1.Functions.of.matrices............................................349
...A.2.M-matrices.......................................................353
...A.3.The.Kronecker.Product............................................354
.......A.3.1.Use.of.Kronecker.Product.for.Solving.Matrix.Equations......357
...A.4.Hamiltonian.Matrices.............................................358
...A.5.Symplectic.Matrices..............................................360
.B.Answers.to.the.Exercises.............................................363
...B.1..Chapter.1.......................................................363
...B.2..Chapter.2.......................................................364
...B.3..Chapter.3.......................................................370
...B.4..Chapter.4.......................................................373
...B.5..Chapter.5.......................................................380
...B.6..Chapter.6.......................................................382
...B.7..Chapter.7.......................................................384
...B.8..Chapter.8.......................................................387
...B.9..Chapter.9.......................................................390
...B.10.Chapter.10......................................................390
...B.11.Chapter.11......................................................391
...B.12.Chapter.12......................................................392
...B.13.Appendix.A......................................................394
.Bibiiography...........................................................399
.Index..................................................................413