Numerical Methods for Stiff Equations and Singular Perturbation Problems (Mathematics and Its Applications)

Approach your problems from It isn't that they can't see the the right end and begin with the solution. It is that they can't see the problem. answers. Then, one day, perhaps you will find the final question. The Hermit Clad in Crane Feathers' G. K. Chesterton, The scandal of in R. Van Gulik's The Chinese Maze Father Brown "The point ofa pin" Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the 'tree' of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non-trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces.

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.