Numerical Methods for Stiff Equations and Singular Perturbation Problems

1. Introduction.- Summary.
- 1.1. Stiffness and Singular Perturbations.
- 1.1.1. Motivation.
- 1.1.2. Stiffness.
- 1.1.3. Singular Perturbations.
- 1.1.4. Applications.
- 1.2. Review of the Classical Linear Multistep Theory.
- 1.2.1. Motivation.
- 1.2.2. The Initial Value Problem.
- 1.2.3. Linear Multistep Operators.
- 1.2.4. Approximate Solutions.
- 1.2.5. Examples of Linear Multistep Methods.
- 1.2.6. Stability, Consistency and Convergence.
- 2. Methods of Absolute Stability.- Summary.
- 2.1. Stiff Systems and A-stability.
- 2.1.1. Motivation.
- 2.1.2. A-stability.
- 2.1.3. Examples of A-stable Methods.
- 2.1.4. Properties of A-stable Methods.
- 2.1.5. A Sufficient Condition for A-stability.
- 2.1.6. Applications.
- 2.2. Notions of Diminished Absolute Stability.
- 2.2.1. A (?)-stability.
- 2.2.2. Properties of A(?)-stable Methods.
- 2.2.3. Stiff Stability.
- 2.3. Solution of the Associated Equations.
- 2.3.1. The Problem.
- 2.3.2. Conjugate Gradients and Dichotomy.
- 2.3.3. Computational Experiments.
- 3. Nonlinear Methods.- Summary.
- 3.1. Interpolatory Methods.
- 3.1.1. Certaine's Method.
- 3.1.2. Jain's Method.
- 3.2. Runge-Kutta Methods and Rosenbrock Methods.
- 3.2.1. Runge-Kutta Methods with v-levels.
- 3.2.2. Determination of the Coefficients.
- 3.2.3. An Example.
- 3.2.4. Semi-explicit Processes and the Method of Rosenbrock.
- 3.2.5. A-stability.- 4 Exponential Fitting.- Summary.
- 4.1. Exponential Fitting for Linear Multistep Methods.
- 4.1.1. Motivation and Examples.
- 4.1.2. Minimax fitting.
- 4.1.3. An Error Analysis for an Exponentially Fitted F1.
- 4.2. Fitting in the Matricial Case.
- 4.2.1. The Matricial Multistep Method.
- 4.2.2. The Error Equation.
- 4.2.3. Solution of the Error Equation.
- 4.2.4. Estimate of the Global Error.
- 4.2.5. Specification of P.
- 4.2.6. Specification of L and R.
- 4.2.7. An Example.
- 4.3. Exponential Fitting in the Oscillatory Case.
- 4.3.1. Failure of the Previous Methods.
- 4.3.2. Aliasing.
- 4.3.3. An Example of Aliasing.
- 4.3.4. Application to Highly Oscillatory Systems.
- 4.4. Fitting in the Case of Partial Differential Equations.
- 4.4.1. The Problem Treated.
- 4.4.2. The Minimization Problem.
- 4.4.3. Highly Oscillatory Data.
- 4.4.4. Systems.
- 4.4.5. Discontinuous Data.
- 4.4.6. Computational Experiments.
- 5. Methods of Boundary Layer Type.- Summary.
- 5.1. The Boundary Layer Numerical Method.
- 5.1.1. The Boundary Layer Formalism.
- 5.1.2. The Numerical Method.
- 5.1.3. An Example.
- 5.2. The ?-independent Method.
- 5.2.1. Derivation of the Method.
- 5.2.2. Computational Experiments.
- 5.3. The Extrapolation Method.
- 5.3.1. Derivation of the Relaxed Equations.
- 5.3.2. Computational Experiments.
- 6. The Highly Oscillatory Problem.- Summary.
- 6.1. A Two-time Method for the Oscillatory Problem.
- 6.1.1. The Model Problem.
- 6.1.2. Numerical Solution Concept.
- 6.1.3. The Two-time Expansion.
- 6.1.4. Formal Expansion Procedure.
- 6.1.5. Existence of the Averages and Estimates of the Remainder.
- 6.1.6. The Numerical Algorithm.
- 6.1.7. Computational Experiments.
- 6.2. Algebraic Methods for the Averaging Process.
- 6.2.1. Algebraic Characterization of Averaging.
- 6.2.2. An Example.
- 6.2.3. Preconditioning.
- 6.3. Accelerated Computation of Averages and an Extrapolation Method.
- 6.3.1. The Multi-time Expansion in the Nonlinear Case.
- 6.3.2. Accelerated Computation of $$\bar f$$.
- 6.3.3. The Extrapolation Method.
- 6.3.4. Computational Experiments: A Linear System.
- 6.3.5. Discussion.
- 6.4. A Method of Averaging.
- 6.4.1. Motivation: Stable Functionals.
- 6.4.2. The Problem Treated.
- 6.4.3. Choice of Functionals.
- 6.4.4. Representers.
- 6.4.5. Local Error and Generalized Moment Conditions.
- 6.4.6. Stability and Global Error Analysis.
- 6.4.7. Examples.
- 6.4.8. Computational Experiments.
- 4.6.9. The Nonlinear Case and the Case of Systems.
- 7. Other Singularly Perturbed Problems.- Summary.
- 7.1. Singularly Perturbed Recurrences.
- 7.1.1. Introduction and Motivation.
- 7.1.2. The Two-time Formalism for Recurrences.
- 7.1.3. The Averaging Procedure.
- 7.1.4. The Linear Case.
- 7.1.5. Additional Applications.
- 7.2. Singularly Perturbed Boundary Value Problems.
- 7.2.1. Introduction.
- 7.2.2. Numerically Exploitable Form of the Connection Theory.
- 7.2.3. Description of the Algorithm.
- 7.2.4. Computational Experiments.- References.