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Numerical Methods for Ordinary Differential Equations

Academic Year 2023/2024 - Teacher: SEBASTIANO BOSCARINO

Expected Learning Outcomes

The goal of this first part of the course is to introduce the student to the computational issues of the solutions of ordinary differential equations (ODEs) and to give several tools for the numerical resolutions of these problems. In particular,some important concepts will be introduced as: consistency, stabilty and convergence of the nuemerical methods  presented during the course. Furthermore some other interesting property of such methods as accuracy and efficiency will be studied. Finally, several Matlab codes regarding the different parts of the course will be presented to the students.

Course Structure

The course of Numerical Analysis is mainly focused on theoretical lessons. During the lessons the theoretical discussion is supported by exercises where applied problems are presented and solved with the help of the computer and by implementing in MatLab of the methods introduced during the lesson.

In case the teaching should be carried out in mixed mode or remotely, it may be necessary to introduce changes with respect to previous statements, in line with the programm planned and outlined in the syllabus.

Learning assessment may also be carried out on line, in case the conditions require it.

Information for students with disabilities and / or SLD

To guarantee equal opportunities and in compliance with the laws in force, interested students can ask for a personal interview in order to plan any compensatory and / or compensatory measures, based on the didactic objectives and specific needs. It is also possible to contact the referent teacher CInAP (Center for Active and Participated Integration - Services for Disabilities and / or SLD) of our Department, prof. Filippo Stanco.

Detailed Course Content

Initial value problems. A brief introduction to ordinary differential equations (ODEs), existence and 

Uniqueness theorem.  Well-posed problem.

 Numerical methods for ODEs:  Explicit and Implicit Euler methods, modified Euler method, Heun method. One-step methods, Taylor methods, Runge-Kutta (RK) methods.

Consistency and convergence of R-K metohds and order conditions. Implicit R-K methods. Variable step size control. Stability analysis for Runge-Kutta methods: stability function, A-stability and L-stability. Stiff problem. Existence and uniqueness of a solution for implicit R-K methods. Collocation methods.

Multistep methods: Adams, Backward Differentiation Formulas (BDF) and linear multistep methods (LMM), predictor and corrector methods, 0-stability, convergence and consistency of LMM.

A brief introduction to Differential Differential-Algebraic Equations (DAEs). Definition of differential  index and special forms of DAEs. Numerical methods for DAEs. Singular perturbation problems. Partitioned and additive Runge-Kutta methods.

Boundary value problems (BVPs). Shooting method. Multiple Shooting method. Finite Difference methods (FD) for BVPs. Consistency, stability and convergence.

Textbook Information

1) G. Naldi, L. Pareschi, G. Russo, Introduzione al calcolo scientifico, McGraw-Hill, 2001.
Testo semplice ed intuitivo. Capitolo 8 è dedicato ai metodi per la risoluzione di ODE.

2) A. Quarteroni, R. Sacco, F. Saleri: Matematica Numerica, Springer Italia, 3° Edizione.
Testo molto ampio e ricco di esempi. Contiene molto materiale e riporta esempi didattici implementati in matlab.

3)V. Comincioli, Analisi Numerica: metodi, modelli, applicazioni, McGraw-Hill, Milano, 1990.
Classico testo di Analilsi Numerica, molto vasto. Contiene molto materiale. Utile strumento di consultazione per alcuni argomenti (es. differenze finite o introduzione ai metodi variazioniali).

4) U. M. Asher e L. R. Petzol, Computer Methods for Ordinary Differential Equations and Differential_Algebraic Equations, Society for Industrial and Applied Mathematics Philadelphia, PA, USA, 1998. Testo utilizzato per la parte riguardante le equazioni differenziali-algebriche.

5) J. Stoer e R. Bulirsch, Introduction to numerical analysis. Ed. Springer Verlag.

6) Ernst Hairer, Gerhard Wanner, Syvert P. Nørsett, Solving ordinary differential equations. I. Nonstiff problems. Third edition, Springer, 2008.

7) Ernst Hairer, Gerhard Wanner, Solving ordinary differential equations. I. Stiff problems. Third edition, Springer, 2010.

Course Planning

 SubjectsText References
1 Runge-Kutta metohds for ODEsBooks: 1) 2) 3) 4) 5) 6) 7)
2Implicit Runge-Kutta methods and collocations methods. Stability and stiff problemsBooks: 4) 5) 6) 7)
3Multistep methods ODEsBooks: 1) 2) 3) 4) 5) 6)
4Differential Algebraic Equations (DAEs)Books: 4) 7)
5Boundary problemsBooks: 3) 4)

Learning Assessment

Learning Assessment Procedures

The  final exam can either be an oral interview or a presentation of a short paper with attached Matlab code on a topic chosen by the student from the course.

The exam is considered passed if an oral interview is judged to be at least sufficient (18/30).

Booking for an exam session is mandatory and must be made exclusively via the internet through the student portal within the set period.

Criteria for assigning marks: both for the written and the oral exams, the following will be taken into account: clarity of presentation, completeness of knowledge, ability to connect different topics. The student must demonstrate that they have acquired sufficient knowledge of the main topics covered during the course and that they are able to carry out at least the simplest of the assigned exercises.

The following criteria will normally be followed to assign the grade:

Not approved: the student has not acquired the basic concepts.

18-23: the student demonstrates minimal mastery of the basic concepts, their skills in exposition and connection of contents are modest.

24-27: the student demonstrates good mastery of the course contents, their presentation and content connection skills are good.

28-30 cum laude: the student has acquired all the contents of the course and is able to explain them fully and connect them with a critical spirit.