34794 - Numerical Analysis (2nd cycle)

Learning outcomes

The student learns the numerical methods for differential problems. At the end of the course the student knows the numerical-mathematical aspects and the main algorithmic methodologies that deal with the numerical solution of differential problems of interest in Engineering.

Course contents

Prerequisites:

A prior knowledge and understanding of Mathematical Analysis, Geometry and Matlab programming is required. Moreover, a prior knowledge of the basic topics of Numerical Analysis is also required.

Program:

1. Numerical Integration: open and closed Gaussian quadrature formulas.

2. Numerical Differentiation: derivative approximation by Finite Differences.

3. Iterative Methods for the numerical solution of linear and non-linear systems.

4. Ordinary Differential Equations: Initial Value Problems. One-step Methods (Runge-Kutta) and Adams Multistep Methods. Predictor-Corrector Methods. BDF Methods. Stiff problems.

5. Boundary-Value Problems for Ordinary Differential Equations: Shooting Methods; Finite Difference Methods; Collocation Methods.

6. Partial Differential Equations. Classification. Hyperbolic, parabolic and elliptic linear problems. Numerical methods for their resolution: Finite Difference methods and Finite Element methods.

Readings/Bibliography

It will be fundamental to use the notes taken during the lectures and the computer material made available on the web. For further study we recommend:

[1] J. Stoer, R. Bulirsch: Introduction to Numerical Analysis (3rd ed.), Springer, 2002.

[2] R.J. LeVeque: Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, Philadelphia, 2007.

[3] C.T. Kelley: Iterative Methods for Linear and Nonlinear Equations, SIAM, 1995.

[4] A. Quarteroni, R. Sacco, F. Saleri: Numerical Mathematics, 2nd ed., Springer, 2007.

[5] U.M. Ascher, L.P.Petzold: Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, 1998.

[6] K. Atkinson, W. Han, D. Stewart: Numerical Solution of Ordinary Differential Equations, John Wiley and Sons, 2009.

[7] D.F. Griffiths, J.W. Dold, D.J. Silvester: Essential Partial Differential Equations, Springer, 2015.

[8] M.S. Gockenbach: Understanding and Implementing the Finite Element Method, SIAM, 2006.

Teaching methods

The course is structured in lectures and exercises in the computer laboratory. More precisely, the lectures on the numerical methods for differential problems described by ordinary or partial differential equations, are followed by laboratory exercises aimed at implementing these methods in MATLAB and developing an adequate sensitivity and awareness of their use.

Assessment methods

The exam aims to verify the achievement of the following educational objectives:

- knowledge of the numerical-mathematical aspects and of the main algorithmic methodologies that deal with the numerical solution of differential problems described by ordinary or partial differential equations;

- ability to solve real problems of interest in engineering by using or developing numerical methods and writing the corresponding algorithms in MATLAB.

The end-of-course exam (the evaluation of which is in thirtieths) will take place in a single test which includes both the development of MATLAB codes for the numerical solution of differential problems, and the written answer to theoretical questions on the topics covered in the lessons.

During the test, the use of support material such as textbooks, notes, computer supports is not allowed.

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PLEASE NOTE: DUE TO THE covid-19 PANDEMIC, THE EXAMINATIONS WILL BE CARRIED OUT ONLINE, WITH THE STUDENTS AT HOME AT WORK ON THEIR PC. IN THIS SITUATION, THE EXAMINATION TEST WILL LAST ONLY 90 MINUTES. TO BE ABLE TO TAKE PART TO THE ONLINE EXAM, STUDENTS WILL HAVE TO INSTALL ON THEIR PC:

1) ZOOM CLIENT FOR MEETINGS (please follow the instructions at https://www.unibo.it/en/services-and-opportunities/online-services/online-services-for-students-1/lessons-and-exams-online
and select “How to take a written exam online via Zoom”)

2) MATLAB

3) A FILE ARCHIVER TO CREATE EITHER .ZIP OR .RAR FILES.

Finally, since during the exam the webcam should be able to capture not only the student's face, but also the table and hands, it would be advisable to use an external webcam (or even an external keyboard) to improve the visibility of the workplace.
The external keyboard could be used in the absence of an external webcam, to move sufficiently far away from the PC and allow the integrated webcam to provide a wider view. In order to use a mobile phone or tablet as an external webcam, it is necessary to install appropriate applications as neither mobile phone nor tablet can be logged in to Zoom. Students must be logged in to Zoom only from the PC where they give the exam.

STUDENTS WHO CANNOT ARRANGE WHAT INDICATED ABOVE ARE INVITED TO CONTACT ME AS SOON AS POSSIBLE

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Teaching tools

The course includes a laboratory activity in which the MATLAB software will be used. The corresponding teaching material will be made available to the student in electronic format and will be downloadable from the IOL platform.

Office hours

See the website of Lucia Romani