OVERVIEW

The course focuses on the numerical approximation of partial differential equations (PDE), also by recalling basic methods for ordinary differential equations (ODE).

Language: Italian

AIMS AND CONTENT

LEARNING OUTCOMES

Survey of numerical methods most used to solve Cauchy problems. Understanding the main issues  in the PDE solution by finite difference methods; ability to implement the corresponding algorithms in relatively straightforward cases, to perform numerical experiments and to interpret the results.

TEACHING METHODS

Teaching style: In presence

The first part of the course (on ODE) is mainly given in the classroom; in the second part (on PDE) classroom lessons (4 hours a week) and lab sessions (2 hours a week) are alternated.

SYLLABUS/CONTENT

Difference equations: the constant-coefficient linear case. Short recall on Runge-Kutta and Multistep methods for initial value Cauchy problems: consistency, convergence, stability, automatic step control. Finite difference approximations of initial and boundary value problems for elliptic, parabolic and hyperbolic PDEs. Explicit and implicit methods. Consistency, stability, convergence. An outline of finite element and finite volume methods. Practical Matlab exercises about the studied methods.

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.

- Afternotes (in italian) authored by P. Fernandes

TEACHERS AND EXAM BOARD

Exam Board

FABIO DI BENEDETTO (President)

PAOLA BRIANZI

CLAUDIO ESTATICO

LESSONS

LESSONS START

The class will start according to the academic calendar.

Class schedule

NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS

EXAMS

EXAM DESCRIPTION

Lab exam, followed by oral exam.

ASSESSMENT METHODS

The lab score takes into account a PDE problem that is assigned in the second part of the course; students must provide a written report on the obtained results, accompanied by comments and Matlab programs. The main purpose of the lab exam is to evaluate the students' ability to apply theory by implementing numerical methods, explaining their behavior and interpreting their results.

The lab score represents a starting basis to decide the final mark, based on the evaluation of the oral exam where the student is asked to expose a rather general topic, showing a good level of understanding and ability of selecting the most relevant aspects.

FURTHER INFORMATION

Previous knowledge:

The course requires basic notions in ODE analysis; It also uses differential calculus tools in multiple variables (such as the Taylor formula). Knowledge of the Euler numerical method can help.

Concerning PDEs, the lessons try to be self-contained; anyway it is useful to have knowledge on basic notions from the analytical or mathematical physics point of view. 

The last part of the course, dedicated to Finite Elements, has functional analysis as a prerequisite.

Attendance is optional.