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.
The entire course will be available online.
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.
|Arieh Iserles||A First Course in the Numerical Analysis of Differential Equations (Edizione 2)||Cambridge University Press||2009||9780521734905|
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 oral, at a distance. A first part dedicated to verifying the understanding of algorithms and basic implementations and a second part dedicated to theory
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