Numerical methods for differential equations
Scientific Disciplinary Sector (SSD)
MAT/08 - NUMERICAL ANALYSIS
Primo semestre dal Oct 4, 2021 al Jan 28, 2022.
The course will discuss, from both the analytic and computational points of view, the main methods for the numerical solution of Ordinary Differential Equations and classical Partial Differential Equations. Exponential Integrators, a current topic of active research in Applied Mathematics, will also be briefly discussed. The course has an important Laboratory component where the methods studied will be implemented using the MATLAB programming platform (using either the official Matlab from Mathworks or else the open source version GNU OCTAVE). At the end of the course the student will be expected to demonstrate that s/he has attained a level of competence in the computational and computer aspects of the course subject, the numerical solution of differential equations.
Prerequisites: linear algebra, differential calculus in one and several variables, integral calculus, basic notions of differential equations, main methods of numerical analysis.
The course will last 52 hours, 20 of which in the computer laboratory.
The course will discuss the following topics:
* Boundary Value Problems: Finite Difference methods, Finite Elements, introduction to Spectral Methods (collocation, discrete Fourier Transform, Galerkin)
* Ordinary Differential Equations: numerical methods for initial value problems, step methods (theta method, variable stepsize Runge-Kutta, introduction to Exponential Integrators) and multistep, stability, absolute stability.
* Partial Differential Equations: basic properties of some of the classical PDEs (Laplace, Heat and Transport), the Method of Lines.
It is expected that there will be a tutor to help with the correction of assigned exercises and with the Laboratory sessions.
Visualizza la bibliografia con Leganto, strumento che il Sistema Bibliotecario mette a disposizione per recuperare i testi in programma d'esame in modo semplice e innovativo.
The purpose of the exam is to see if the student is able to recall and produce the theory of numerical methods for differential equations presented during the lectures and Laboratory and knows how to use Computer resources for possible further investigation. Moreover, the student must show that s/he knows how to program in the specific software introduced during the course. The exam method is both written (solution with the computer of given exercises within a certain amount of time, with the possibility to use any material) and oral (devoted to the theory). To access the oral part it is mandatory to succeed in the written one.
The examination procedure could undergo variations depending on the evolution of the situation.