NUMERICAL METHODS

This course is one of the characterizing educational activities of the Master's Degree course and allows the student to acquire the knowledge and understanding the fundamental and applicative concepts of numerical analysis

The aim of the course is to provide basic knowledge of numerical analysis for linear algebra problems, numerical integration and differential equations

At the end of the course, the student will be able to approximate various problems of a mathematical nature and to choose the best algorithm for the problem addressed.

Knowledge and understanding
At the end of the course, students will have acquired notions and results related to methods for the numerical solution of linear systems and eigenvalue problems and for the discretization of differential equations and will have acquired techniques related to the implementation of algorithms for the efficient solution of the problems treated.

Ability to apply knowledge and understanding
Students who have passed the exam will be able to use the methodologies illustrated in the Course for the numerical solution of a linear system or of an eigenvalue problem and for the discretization of differential equations, and will be able to predict its performance depending on the characteristics of the problem to be treated.

Autonomy of judgment
Students who have passed the exam will be able to choose the most suitable algorithms for the solution of the problem considered from all the algorithms they have studied in the course, having also acquired the tools to make the changes that may be necessary to improve its performance.

Communication skills
Students will have developed the ability to present the concepts, ideas and methodologies covered in the course.

Learning ability
The acquired knowledge will allow students who have passed the exam to tackle the study, individually or in a Master's Degree course, of more specialized aspects of numerical linear algebra and numerical modeling for differential problems, being able to understand the specific terminology and identify the most relevant issues.

Mathematical analysis I, Linear Algebra, Informatics I, Mathematical analysis II

Numerical linear algebra
Review of matrix factorization and direct methods for the solution of linear systems. Least squares problem. Band and block structure systems. Gradient methods. Krylov's methods. Review of the methods for the computation of the eigenvalues. Decomposition to singular values.

Numerical methods for ordinary differential equations
Cauchy's problem. Recalls on one-step methods. Consistency, zero-stability and convergence. Absolute stability. Multi-step and variable-step methods. Notes on the limit problem and finite difference method.

Numerical methods for linear partial differential equations
Two-dimensional approximation techniques based on finite differences. Poisson equation: 5 and 9 point methods. The Cauchy problem for the scalar transport equation. Numerical schemes: Euler forward, Lax-Friedrichs, Lax-Wendroff, Upwind, Euler backward. Consistency, stability and convergence. Approximation of hyperbolic systems and wave equation. Leap-frog scheme. Equation of heat. Numerical solution: the method and the Crank Nicolson scheme.

The course includes laboratory activities for the development of MATLAB codes for the numerical solution of linear algebra problems and for the discretization of differential problems.

A. Quarteroni, R. Sacco, F. Saleri, P. Gervasio, “Numerical mathematics”, Springer,
RJ Leveque, "Finite Difference methods for ordinary and partial differential equations", SIAM book, 2007

How to verify what the students learned

The exam consisting in the discussion of the methodologies illustrated in the course and its implementation in MATLAB.

The student must demonstrate that he has acquired a knowledge of the most relevant topics and methods illustrated in the course and must be able to discuss their implementation in MATLAB.

Standard classes (75%) - Exercises and MATLAB (25%)

English

Oral exam

oral