ChapterIV.StiffProblems-One-StepMethods.
IV.IExamplesofStiffEquations
ChemicalReactionSystems
ElectricalCircuits
Diffusion
A"Stiff"Beam
HighOscillatiom
Exercises
IV.2StabilityAnalysisforExplicitRKMethods
StabilityAnalysisforEuler'sMethod
ExplicitRunge-KuttaMethods
ExtrapolationMethods
AnalysisoftheExamplesofIV.I
AutomaticStiffnessDetection
Step-ControlStability
APIStepSizeControl
StabilizedExplicitRunge-KuttaMethods
Exercises
IV.3StabfiityFunctionofImplicitRK-Methods
TheStabilityFunction
A-Stability
L-StabilityandA(a)-Stability
NumericalResults
StabilityFunctionsofOrders
Pad6ApproximationstotheExponentialFunction
Exercises
IV.4OrderStars
Introduction
OrderandStabilityforRationalApproximations
StabilityofPad6Approximations
ComparingStabilityDomains
RationalApproximationswithRealPoles
TheReal-PoleSandwich
MultipleReal-PoleApproximations
Exercises
IV.5ConstructionofImplicitRunge-KuttaMethods.
GaussMethods
RadauIAandRadauHAMethods
LobattoIIIA,IIIBandmCMethods
TheW-Transformation
ConstructionofImplicitRunge-KuttaMethods
StabilityFunction
PositiveFunctions
Exercises
IV.6DiagonallyImplicitRKMethods
OrderConditions
StifflyAccurateSDIRKMethods
TheStabilityFunction
MultipleReal-PoleApproximationswithR(oo)=0
ChoiceofMethod
Exercises
IV.7Rosenbrock-TypeMethods
DerivationoftheMethod
OrderConditions
TheStabilityFunction
ConstructionofMethodsofOrder4
HigherOrderMethods
ImplementationofRosenbrock-TypeMethods
The"Hump"
MethodswithInexactJacobian(W-Methods)
Exercises
IV.8ImplementationofImplicitRunge-KuttaMethods
ReformulationoftheNonlinearSystem
SimplifiedNewtonIterations
TheLinearSystem
StepSizeSelection
ImplicitDifferentialEquations
AnSDIRK-Code
SIRK-Methods
Exercises
IV.9ExtrapolationMethods
ExtrapolationofSymmetricMethods
Smoothing
TheLinearlyImplicitMid-PointRule
ImplicitandLinearlyImplicitEulerMethod
Implementation
Exercises
IV.10NumericalExperiments
TheCodesUsed
TwelveTestProblems
ResultsandDiscussion
PartitioningandProjectionMethods
Exercises
IV.11ContractivityforLinearProblems
EuclideanNorms(TheoremofyonNeumann)
ErrorGrowthFunctionforLinearProblems
SmallNonlinearPerturbations
ContractivityinI1'Iiooand
StudyoftheThresholdFactor
AbsolutelyMonotonicFunctions
Exercises
IV.12B-StabilityandContractivity
One-SidedLipschitzCondition
B-StabilityandAlgebraicStability
SomeAlgebraicallyStableIRKMethods
AN-Stability
ReducibleRunge-KuttaMethods
TheFqlnivalenceTheoremforS-IrreducibleMethods
ErrorGrowthFunction
ComputationofB(x)
Exercises
IV.13PositiveQuadratureFormulasandB-StableRK-Methods
QuadratureFormulasandRelatedContinuedFractions
NumberofPositiveWeights
CharacterizationofPositiveQuadratureFormulas
NecessaryConditionsforAlgebraicStability
CharacterizationofAlgebraicallyStableMethods
The"Equivalence"ofA-andB-Stability
Exercises
IV.14ExistenceandUniquenessofIRKSolutions
Existence
ACounterexample
InfluenceofPerturbationsandUniqueness
Computationofao(A-1)
MethodswithSingularA
LobattoIIICMethods
Exercises..
IV.15B-Convergence
TheOrderReductionPhenomenon
TheLocalError
ErrorPropagation
B-ConvergenceforVariableStepSizes
B-ConvergenceImpliesAlgebraicStability
TheTrapezoidalRule
orderReductionforRosenbrockMethods
Exercises
ChapterV.Mu!fistepMethodsforStiffProblems
V.1StabilityofMultistepMethods
TheStabilityRegion
AdamsMethods
Predictor-CorrectorSchemes
Nystr6mMethods
BDF
TheSecondDaldquistBarrier
Exercises
V.2"Nearly"A-StableMultistepMethods
A(a)-StabilityandStiffStability
HighOrderA(a)-StableMethods
ApproximatingLoworderMethodswithHighOrderOnes
ADiscTheorem
AccuracyBarriersforLinearMultistepMethods
Exercises..
V.3GeneralizedMultistepMethods
SecondDerivativeMultistepMethodsofEnright
SecondDerivativeBDFMethods
BlendedMulfistepMethods
ExtendedMultistepMethodsofCash
MulfistepCollocationMethods
Methodsof"Radau"Type
Exercises
V.4OrderStarsonRiemannSurfaces
RiemannSurfaces
PolesRepresentingNumericalWork
OrderandOrderStars
The"DanielandMooreConjecture"
MethodswithPropertyC
GeneralLinearMethods
DualOrderStars
Exercises
V.5ExperimentswithMultistepCodes
TheCodesUsed
Exercises
V.6One-LegMethodsandG-Stability
One-Leg(Multistep)Methods
ExistenceandUniqueness
G-Stability
AnAlgebraicCriterion
TheEquivalenceofA-StabilityandG-Stability
ACriterionforPositiveFunctions
ErrorBoundsforOne-LegMethods
ConvergenceofA-StableMultistepMethods
Exercises
V.7ConvergenceforLinearProblems
DifferenceEquationsfortheGlobalError
TheKreissMatrixTheorem
SomeApplicationsoftheKreissMatrixTheorem
GlobalErrorforProtheroandRobinsonProblem
ConvergenceforLinearSystemswithConstantCoefficients
MatrixValuedTheoremofvonNeumann
DiscreteVariationofConstantsFormula
Exercises
V.8ConvergenceforNonlinearProblems
ProblemsSatisfyingaOne-SidedLipschitzCondition
MultiplierTechnique
MultipliersandNonlinearities
DiscreteVariationofConstantsandPerturbations
ConvergenceforNonlinearParabolicProblems
Exercises
V.9AlgebraicStabilityofGeneralLinearMethods
G-Stability
AlgebraicStability
AN-StabilityandEquivalenceResults
MultistepRunge~KuttaMethods
SimplifyingAssumptions
QuadratureFormulas
AlgebraicallyStableMethodsofOrder2s
B-Convergence
Exercises
ChapterVI.SingularPerturbationProblems
andIndex1Problems
VI.1SolvingIndex1Problems
AsymptoticSolutionofvanderPol'sEquation
Thee-EmbeddingMethodforProblemsofIndex1
StateSpaceFormMethod
ATransistorAmplifier
ProblemsoftheFormMu=(u)
ConvergenceofRunge-KuttaMethods
Exercises
VI.2MultistepMethods
MethodsforIndex1Problems
ConvergenceforSingu/arPerturbationProblems
Exercises
VI.3EpsilonExpansionsforExactandRKSolutions
ExpansionoftheSmoothSolution
ExpansionswithBoundaryLayerTerms
EstimationoftheRemainder
ExpansionoftheRunge-KuttaSolution
ConvergenceofRK-MethodsforDifferential-AlgebraicSystems
ExistenceandUniquenessoftheRunge-KuttaSolution
InfluenceofPerturbations
EstimationoftheRemainderintheNumericalSolution
NumericalConfirmation
PerturbedInitialValues
Exercises
VIARosenbrockMethods
DefinitionoftheMethod
DerivativesoftheExactSolution
TreesandElementaryDifferentials
TaylorExpansionoftheExactSolution
TaylorExpansionoftheNumericalSolution
OrderConditions
Convergence
StifflyAccurateRosenbrockMethods
ConstructionofRODAS,aStifflyAccurateEmbeddedMethod
InconsistentInitialValues
Exercises
VI.5ExtrapolationMethods
LinearlyImplicitEulerDiscrefization
PerturbedAsymptoticExpansion
OrderTableau
ErrorExpansionforSingularPerturbationProblems
DenseOutput
Exercises
VI.6QuasilinearProblems
Example:MovingFiniteElements
ProblemsofIndexOne
C(u)u
NumericalTreatmentof(y)y=f(y)
ExtrapolationMethods
Exercises
ChapterVII.Differential-AlgebraicEquations
ofHigherIndex
Vii.1TheIndexandVariousExamples
LinearEquationswithConstantCoefficients
DifferentiationIndex
DifferentialEquationsonManifolds
ThePerturbationIndex
ControlProblems
MechanicalSystems
Exercises
VH.2IndexReductionMethods
IndexReductionbyDifferentiation
StabilizationbyProjection
DifferentialEquationswithInvariants
MethodsBasedonLocalStateSpaceForms
OverdeterminedDifferential-AlgebraicEquations
UnstructuredHigher'IndexProblems
Exercises
VH.3MultistepMethodsforIndex2DAlE
ExistenceandUniquenessofNumericalSolution.
InfluenceofPerturbations
TheLocalError
ConvergenceforBDF
GeneralMultistepMethods
SolutionoftheNonlinearSystembySimplifiedNewton
Exercises
VIL4Runge-KuttaMethodsforIndex2DAE
TheNonlinearSystem
EstimationoftheLocalError
Convergenceforthey-Component
Convergenceforthez-Component
CollocationMethods
SuperconvergenceofCollocationMethods
ProjectedRunge-KuttaMethods
SummaryofConvergenceResults
Exercises
VII.SOrderConditionsforIndex2DAlE
DerivativesoftheExactSolution
TreesandElementaryDifferentials
TaylorExpansionoftheExactSolution
DerivativesoftheNumericalSolution
OrderConditions
SimplifyingAssumptions
ProjectedRunge-KuttaMethods
Exercises
VIL6Half-ExplicitMethodsforIndex2Systems
Haft-ExplicitRunge-KuttaMethods
ExtrapolationMethods
/3-BlockedMultistepMethods
Exercises
VIL7ComputationofMultibodyMechanisms
DescriptionoftheModel
FortranSubroutines
ComputationofConsistentInitialValues
NumericalComputations
AStiffMechanicalSystem
Exercises
VI/.8SymplecficMethodsforConstrainedHamUtonianSystems
PropertiesoftheExactFlow
FirstOrderSymplecticMethod
SHAKEandRATILE
TheLobattoIIIA-IIIBPair
CompositionMethods
BackwardErrorAnalysis(forODEs)
BackwardErrorAnalysisonManifolds
Exercises
Appendix.FortranCodes
DriverfortheCodeRADAU5
SubroutineRADAU5
SubroutineRADAUP
SubroutineRODAS
SubroutineSEULEX
ProblemswithSpecialStructure
UseofSOLOUTandofDenseOutput
Bibliography
SymbolIndex
SubjectIndex...
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