04524 - Numerical Analysis

Learning outcomes

At the end of the course, students thoroughly know the theoretical and computational properties of the principal numerical methods for initial value ordinary differential equations, and of some advanced numerical methods for partial differential equations. In particular, students are able to analyze the theoretical properties of numerical methods and to critically examine the computational results.

Course contents

• Iterative methods for systems of nonlinear equations: the method of successive approximations (convergence properties, a-posteriori and a-priori error estimates, computational aspects); Newton’s method (local convergence theorem, Kantorovich theorem), inexact Newton method; brief notes on the modified Newton method (global convergence, Zoutendijk’s lemma).
• Ordinary differential equations: the initial value problem. One-step and multistep numerical methods, consistency, zero-stability and convergence; absolute stability. Stiff problems: characteristics and numerical solution.
• Ordinary differential equations: the boundary value problem. Linear and nonlinear second-order equations; shooting method and finite difference method; consistency, stability, and convergence.
• Partial differential equations (overview). Diffusion equations: method of lines and ADI (alternating direction implicit) methods; stability and convergence. Transport equations: method of lines, Lax–Wendroff method; stability and convergence.

The course includes a laboratory activity where students can practice and experiment with the proposed computational methods.

The basic courses of the Bachelor's Degree in Mathematics are the necessary prerequisites to successfully attend the course.

Readings/Bibliography

R.J. LeVque, Finite Difference Methods for Ordinary and Partial Differential Equations, SIAM, 2007

D.F. Griffths and D.J. Higham. Numerical Methods for Ordinary Differential Equations: Initial Value Problems. Springer, 2010.

J. Nocedal, S. J. Wright. Numerical Optimization. Springer, 2006.

Teaching methods

Classroom lectures for the theory, and computer Laboratory for codes implementation and analysis of proposed examples.

In consideration of the type of activity and teaching methods adopted, the frequency of this training activity requires the prior participation of all students in Modules 1 and 2 of safety training in the places of study, in e-learning mode.

Assessment methods

The exam consists of an oral test on the topics covered in class and on the exercises carried out in the calculation laboratory.

The oral test is structured to verify the knowledge and level of understanding of the topics covered during the course.

Particular attention is given to both the ability to communicate the subject critically and the use of appropriate mathematical language.

Teaching tools

Slides provided by the teacher. The teaching material will be available on the University of Bologna e-learning platform (https://virtuale.unibo.it).

Office hours

See the website of Germana Landi