Scientific Disciplinary Sector (SSD)
MAT/08 - NUMERICAL ANALYSIS
Primo semestre dal Oct 3, 2022 al Jan 27, 2023.
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 and basic notions
Linear algebra, differential calculus in one and several variables, integral calculus, basic notions of differential equations, main methods of numerical analysis.
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.
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The course will last 52 hours in presence, 20 of which in the computer laboratory.
Learning assessment procedures
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.
To pass the exam, you will have to demonstrate:
* to know and have understood the fundamental numerical methods for initial value problems
* to know and to have understood the fundamental numerical methods for boundary value problems
* to know and to have understood the fundamental numerical methods for some simple partial differential equations
* to know and to have understood the properties of the main numerical methods
* having an adequate capacity for analysis and synthesis and abstraction
* knowing how to apply this knowledge to solve problems and exercises, knowing how to argue your reasoning with mathematical rigor.
Criteria for the composition of the final grade
The final mark is given by the arithmetic mean of the mark of the written and of the oral mark.