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CODE 61473 2022/2023 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA 6 cfu anno 1 MATEMATICA 9011 (LM-40) - GENOVA MAT/08 GENOVA 2° Semester Questo insegnamento è un modulo di: AULAWEB

## 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 intends to introduce the main problems that must be faced in the numerical solution of PDE, also with reference to the implementation of the corresponding algorithms and to the interpretation of the results for the relative 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.

### 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).

### 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)

FEDERICO BENVENUTO

CLAUDIO ESTATICO (President Substitute)

ALBERTO SORRENTINO (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 with DSA certification ("specific learning disabilities"), disability or other special educational needs are advised to contact the teacher at the beginning of the course to agree on teaching and examination methods that, in compliance with the teaching objectives, take account of individual learning arrangements and provide appropriate compensatory tools.

### 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.