OVERVIEW

After a brief recall on numerical methods for ordinary differential equations (ODE), the course provides the basic concepts on the numerical approximation of partial differential equations (PDE).

Lessons are given in Italian.

AIMS AND CONTENT

LEARNING OUTCOMES

The course aims to introduce the main issues to be faced in the numerical solution of PDE, including implementation of the corresponding algorithms and interpretation of the results for the related numerical experiments.

AIMS AND LEARNING OUTCOMES

At the end of the course, the student will be able:

• to understand the main finite difference discretization methods for the different classes of PDEs;
• to implement these methods even on non-trivial examples;
• to evaluate their performance according to the choice of parameters;
• to develop soft skills at an advanced level..

PREREQUISITES

The course is based on the analytical and numerical notionsabout Ordinary Differential Equations (ODE), respectively developed in the previous courses Mathematical Analysis 2 and Foundations of Numerical Analysis; it also uses differential calculus tools in several variables (for example the Taylor formula) introduced in the second year analysis courses.

Regarding Partial Differential Equations (PDEs), the lessons try to be self-contained; it is however useful that the student has in his own curriculum Differential Equations 1 and / or Models of Continuous Systems and Applications.

The laboratory part requires a good familiarity with the Matlab language.

TEACHING METHODS

Traditional method.

After the first weeks, lessons are partly given in the classroom (3 hours per week) and partly given in the laboratory (2 hours per week, which are increased in the last part of the semester).

In addition, innovative teaching tools will be proposed (in particular, reflective activities on self evaluation) allowing students to acquire soft skills.

SYLLABUS/CONTENT

Review of Runge-Kutta and Multistep methods for initial value Cauchy problems: consistency, convergence, stability, automatic step control. Finite difference approximations of initial and / or boundary value problems for elliptic, parabolic and hyperbolic PDEs. Explicit and implicit methods. Consistency, stability, convergence. Outline of finite element methods. Laboratory exercises in Matlab on the methods studied.

RECOMMENDED READING/BIBLIOGRAPHY

- J. D. Lambert, Computational Methods in Ordinary Differential Equations. John Wiley & Sons, London, 1973.
- J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations. Second Edition, SIAM Publications, 2004.

- Quarteroni, A.; Valli, A., Numerical Approximation of Partial Differential Equations. Berlin etc., Springer-Verlag 1994.
 

TEACHERS AND EXAM BOARD

Exam Board

FABIO DI BENEDETTO (President)

CLAUDIO ESTATICO

FEDERICO BENVENUTO (President Substitute)

PAOLA FERRARI (President Substitute)

LESSONS

LESSONS START

According to the academic calendar.

Class schedule

The timetable for this course is available here: Portale EasyAcademy

EXAMS

EXAM DESCRIPTION

The final exam grade takes into account the laboratory grade and the oral exam.

Students who have valid certification of physical or learning disabilities on file with the University and who wish to discuss possible accommodations or other circumstances regarding lectures, coursework and exams, should speak both with the instructor and with Professor Sergio Di Domizio (sergio.didomizio@unige.it), the Department’s disability liaison.

ASSESSMENT METHODS

The laboratory is a group activity. The evaluation takes into account a project concerning PDEs that is assigned (possibly in a "personalized" way) around the third week, and requires a written report on the results obtained, accompanied by comments and Matlab programs. There is no mandatory deadline for deliveries.

The main purpose of the test is to evaluate the students' ability implement the numerical methods in computer programs, to explain their behavior and to interpret the results by applying the theory developed.

The evaluation (out of thirty) is usually communicated 7-10 days after delivery and is definitive. It is therefore not allowed to repeat the test; unless otherwise decided by the teachers (communicated in time), the grade never expires.

In the oral exam the degree of understanding of the subject is assessed, as well as the ability to present and connect the various concepts.

Exam schedule