.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

.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

.......5.3.1.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

.......6.3.1.The.Case.of.a.Nonuniform.Mesh..............................150

.......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

.......7.3.3.Unsymmetric.ETR2s..........................................170

...7.4.Top.Order.Methods.(TOMs).........................................171

.......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

.......8.5.1.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

....11.5.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.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