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NUMERICAL METHODS FOR DIFFERENTIAL EQUATIONS - 7 CFU

Teacher

Mario Putti

Scheduled Period

I Year - 2 Semester | 02/03/2020 - 12/06/2020

Hours: 64 (16 laboratorio, 48 lezione)

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Prerequisites

Mathematical Analysis 1 and 2, with elements of Differential Equations and functional analysis. Numerical Analysis and linear algebra. The lab projects require some knowledge of Matlab programming.

Target skills and knowledge

The course deals with methods of scientific computing and numerical analysis for the solution of partial differential equations. We will address both application and implementation issues as well as theoretical results. The course will also address many of the instruments that are necessary to complete the numerical solution of a PDE, such as solution of ODEs, solution of large sparse linear systems of equations. The lab projects will provide the students with the opportunity to challenge themselves in practical implementation issues.

Examination methods

Oral examination with discussion on the lab projects.

Assessment criteria

30% lab projects
70% oral discussion

Course contents

Ordinary differential equations. Generalities, existence and uniqueness. Discrete methods: one step-methods, Runge-Kutta methods. stability and convergemce. Multi-step methods. Stiff problems, linear and nonlinear stability, implementation.
Partial differential equations: characterization with description of most important model problems. FEM methods for elliptic equations: variational formulation, Hilbert spaces; boundary conditions (Dirichlet, Neumann, Cauchy). Abstract FEM formulation: energy norm, discretization, error estimates, regularity of the solution. Parabolic equations: spatio-temporal discretizations. Error and stability estimates for Euler and Crank-Nicolson methods. Applications to nonlinear problems.

Planned learning activities and teaching methods

Classroom and computer laboratory. The theoretical notions will be discussed on the blackboard. The implementation issues and usage of the different algorithms will be discussed in the computer lab.

Additional notes about suggested reading

Lecture notes written by the teacher will be available for most of the material.

Textbooks (and optional supplementary readings)

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