This book is focused on the numerical discretization of ordinary differential equations (ODEs), under several perspectives. The attention is first conveyed to providing accurate numerical solutions of deterministic problems. Then, the presentation moves to a more modern vision of numerical approximation, oriented to reproducing qualitative properties of the continuous problem along the discretized dynamics over long times. The book finally performs some steps in the direction of stochastic differential equations (SDEs), with the intention of offering useful tools to generalize the techniques introduced for the numerical approximation of ODEs to the stochastic case, as well as of presenting numerical issues natively introduced for SDEs.

The book is the result of an intense teaching experience as well as of the research carried out in the last decade by the author. It is both intended for students and instructors: for the students, this book is comprehensive and rather self-contained; for the instructors, there is material for one or more monographic courses on ODEs and related topics. In this respect, the book can be followed in its designed path and includes motivational aspects, historical background, examples and a software programs, implemented in Matlab, that can be useful for the laboratory part of a course on numerical ODEs/SDEs.

The book also contains the portraits of several pioneers in the numerical discretization of differential problems, useful to provide a framework to understand their contributes in the presented fields. Last, but not least, rigor joins readability in the book.

Raffaele D'Ambrosio is Full Professor of Numerical Analysis at the University of L'Aquila. He got his Ph.D. in Mathematics in 2006, by a joint program between the University of Salerno and Arizona State University. In 2011 he has been awarded with Galileo Galilei Prize for young researchers. In 2014 he has been Fulbright Research Scholar at Georgia Institute of Technology. His main research interests regard structure-preserving approximation of deterministic and stochastic evolutive problems.

1 Ordinary differential equations

1.1 Initial value problems

1.2 Well-posedness

1.3 Discontinuous ODEs

1.4 Dissipative problems

1.5 Conservative problems

1.6 Stability of solutions

1.7 Exercises

2 Discretization of the problem

2.1 Domain discretization

2.2 Difference equations: the discrete counterpart of differential equations

2.2.1 Linear difference equations

2.2.2 Homogeneous case

2.2.3 Inhomogeneous case

2.3 Step-by-step schemes

2.4 A theory of one-step methods

2.4.1 Consistency

2.4.2 Zero-stability

2.4.3 Convergence

2.5 Handling implicitness

2.6 Exercises

3 Linear Multistep Methods

3.1 The principle of multistep numerical integration

3.2 Handling implicitness by fixed point iterations

3.3 Consistency and order conditions

3.4 Zero-stability

3.5 Convergence

3.6 Exercises

4 Runge-Kutta methods

4.1 Genesis and formulation of Runge-Kutta methods

4.2 Butcher theory of order

4.2.1 Rooted trees

4.2.2 Elementary differentials

4.2.3 B-series

4.2.4 Elementary weights

4.2.5 Order conditions

4.3 Explicit methods

4.4 Fully implicit methods

4.4.1 Gauss methods

4.4.2 Radau methods

4.4.3 Lobatto methods

4.5 Collocation methods

4.6 Exercises

5 Multivalue methods

5.1 Multivalue numerical dynamics

5.2 General linear methods representation

5.3 Convergence analysis

5.4 Two-step Runge-Kutta Methods

5.5 Dense output multivalue methods

5.6 Exercises

6 Linear stability

6.1 Dahlquist test equation

6.2 Absolute stability of linear multistep methods

6.3 Absolute stability of Runge-Kutta methods

6.4 Absolute stability of multivalue methods

6.5 Boundary locus

6.6 Unbounded stability regions

6.6.1 A-stability

6.6.2 Padé approximations

6.6.3 L-stability

6.7 Order stars

6.8 Exercises

7 Stiff problems

7.1 Looking for a definition

7.2 Prothero-Robinson analysis

7.3 Order reduction of Runge-Kutta methods

7.4 Discretizations free from order reduction

7.4.1 Two-step collocation methods

7.4.2 Almost collocation methods

7.4.3 Multivalue collocation methods free from order reduction

7.5 Stiffly-stable methods: backward differentiation formulae

7.6 Principles of adaptive integration

7.6.1 Predictor-corrector schemes

7.6.2 Stepsize control strategies

7.6.3 Error estimation for Runge-Kutta methods

7.6.4 Newton iterations for fully implicit Runge-Kutta methods

7.7 Exercises

8 Geometric numerical integration

8.1 Historical overview

8.2 Principles of nonlinear stability for Runge-Kutta methods

8.3 Preservation of linear and quadratic invariants

8.4 Symplectic methods

8.5 Symmetric methods

8.6 Backward error analysis

8.6.1 Modified differential equations

8.6.2 Truncated modified differential equations

8.6.3 Long-term analysis of symplectic methods

8.7 Long-term analysis of multivalue methods

8.7.1 Modified differential equations

8.7.2 Bounds on the parasitic components

8.7.3 Long-time conservation for Hamiltonian systems

8.8 Exercises

9 Numerical methods for stochastic differential equations

9.1 Discretization of the Brownian motion

9.2 Ito and Stratonovich integrals

9.3 Stochastic differential equations

9.4 One-step methods

9.4.1 Euler-Maruyama and Milstein methods

9.4.2 Stochastic ¿-methods

9.4.3 Stochastic perturbation of Runge-Kutta methods

9.5 Accuracy analysis

9.6 Linear stability analysis

9.6.1 Mean-square stability

9.6.2 Mean-square stability of stochastic ¿-methods

9.6.3 A-stability preserving SRK methods

9.7 Principles of stochastic geometric numerical integration

9.7.1 Nonlinear stability analysis: exponential mean-square contractivity

9.7.2 Mean-square contractivity of stochastic ¿-methods

9.7.3 Nonlinear stability of stochastic Runge-Kutta methods

9.7.4 A glance to the numerics for stochastic Hamiltonian problems

9.8 Exercises

A Summary of test problems

Bibliography