Numerical Methods for Ordinary Differential Equations

Academic Year 2024/2025 - 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.

Dublin Indicators:

1) Knowledge and Understanding. The course represents an introductory step into the advanced study of numerical methods for ordinary differential equations and addresses topics of significant interest for applications. It builds upon foundational knowledge in mathematical analysis, programming, and numerical computation, enhancing these basic skills while introducing students to algorithmic structures and computational procedures using MATLAB as a specific programming language. The use of multiple textbooks, along with some course notes prepared by the instructor, aims to improve students' reading and comprehension skills. Some practical results will be presented using MATLAB software during the course, aiming to enhance the capacity for applied and numerical problem-solving and to provide advanced computational skills.

2) Applying Knowledge and Understanding. The use of MATLAB software during the course aims to enhance the ability to solve both theoretical and practical problems, improve the mastery of programming concepts and scientific computing skills, and foster problem-solving abilities. During the course, some computational verifications of theoretical results (such as the accuracy and stability of a numerical method) will be presented to illustrate certain theorems. Additionally, these results enable students to use computational and IT skills to study mathematical problems.

3) Making Judgments. As this is an introductory course, students are expected to improve their logical reasoning skills and recognize the importance of hypotheses in reaching conclusions. The various computational exercises presented and conducted in class, later reviewed by the student or discussed with classmates, are often approached using different logical frameworks. This helps develop critical thinking skills in students, who learn to follow not necessarily the most immediate or least efficient path, but rather the one that is most appropriate from a computational perspective. In-class exercises promote group work in addition to individual study. The extensive literature recommended encourages individual initiative in further study, which is a foundational step toward achieving autonomy in addressing new problems in applied mathematics.

4) Communication Skills. When faced with a real-world problem of an industrial or financial nature, represented by relatively elementary situations of practical interest, students, through their mathematical training and especially their numerical modeling skills, are first encouraged to explain the motivations behind the problem, depending on the context in which it originated, and then effectively communicate the actual solution of the problem. The various texts suggested for the course, some in English, help students become familiar with the scientific terminology of numerical computation expressed in English and train them in using English for scientific communication. The oral examination or the presentation of a report requires students to clearly and rigorously express the various topics covered in the course.

5) Learning Skills. The learning required for this course serves as an initial step toward developing a flexible mindset, which is useful for further studies within the Master's degree program or for entering various professional fields. Moreover, familiarity with the MATLAB language provides a computational tool that can be independently utilized as an IT aid in Master’s degree courses as well as in the workplace. Additionally, the great flexibility of the MATLAB scientific software will enable students to quickly adapt to the evolution of IT tools and to maintain their scientific skills.

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.

Required Prerequisites

A knowledge of differential and integral calculus for functions of one or more variables, complex numbers, vector and matrix calculus, programming concepts and familiarity with the MATLAB language, as well as numerical computation methods, is assumed.

Attendance of Lessons

Attendance is not mandatory for passing the exam, but it is strongly recommended.

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. Semi-implicit 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

	Subjects	Text References
1	Runge-Kutta methods for ODEs	Books: 1) 2) 3) 4) 5) 6) 7)
2	Implcit and collocations Metodi per ODEs. Stabilty and Stiff problems	Books: 4) 5) 6) 7)
3	Multistep methods for ODEs	Books: 1) 2) 3) 4) 5) 6)
4	Differential Algegbraic Equations (DAEs)	Books: 4) 7)
5	Boundary problems	Books: 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.

Examples of frequently asked questions and / or exercises

Theorem of Convergence for the Explicit Euler Method

Discuss the stability of Runge-Kutta methods.

Degree Course in

Mathematics