Learning outcomes

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.

Program

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.

Examination Methods

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 will consist of two parts. The first part will be held in a Laboratory where the student will be given two hours to individually implement the numerical methods necessary for the solution of the assigned questions. These questions will be based on finite difference methods with fixed stepsize for Boundary Value Problems, fixed stepsize methods for initial value problems and the Method of Lines for Partial Differential Equations. A pass will be given for a mark of 15/30 or higher. To be admitted to the second part of the exam, the oral, it is required to have first passed the written part. Marks for the written part will remain valid until, and not after, the beginning of the following semester. The oral exam will be based on all the material presented during the course, with the exception of the details of the Discrete Fourier Transform. The final course mark will be the average of the marks for the two parts of the exam.