35433 - Numerical Methods

Learning outcomes

At the end of the course, students know basic numerical methods for evolutive ordinary and partial differential problems, together with their main theoretical and computational properties. In particular, students are able to analyze the properties of numerical methods; constructively examine corresponding computational results; advance their scientific computing education in higher level courses; employ the acquired numerical skills in a variety of application areas.

Course contents

Main topics:

• Numerical solution of Ordinary Differential Equations (ODEs): Initial Value Problems
  • First order equations and systems:
    • Onestep-multistep methods.
    • Convergence and Stability.
• Numerical solution of ODEs: Boundary Value Problems
  • Shooting Method
  • Finite difference methods
  • Galerkin’s Method

Related topics:

• Nonlinear systems
• Gaussian Quadrature Formulas

Prerequisites

- Matlab programming

- Floating point arithmetic.
- Numerical methods for the solution of linear systems;

- Numerical methods for the solution of nonlinear equations.
- Data approximation: polynomial and piecewise polynomial functions; interpolation and least-squares approximation.
- Numerical integration: Newton-Cotes quadrature formulas.

Readings/Bibliography

• Course Lecture notes

• U. Ascher and L. Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. SIAM, 1998.
• D.F. Griffths and D.J. Higham. Numerical Methods for Ordinary Differential Equations: Initial Value Problems. Springer, 2010.
• Randal J. LeVeque. Finite Difference Methods for Ordinary and Partial Differential Equations. SIAM, 2007.
• Alfio Quarteroni, Riccardo Sacco, and Fausto Saleri. Numerical Mathematics (Texts in Applied Mathematics). Springer-Verlag, Berlin, Heidelberg, 2006.
• H.B.Keller. Numerical Methods for Two-Point Boundary Value Problems. Dover Ed., 2018.

Teaching methods

Classroom lectures: the numerical methods are introduced, and the theoretical properties assessed.

Computer laboratory-guided lectures: The numerical methods are developed and analyzed through examples reported in laboratory assignment sheets.

During the course, several topics will be proposed as subjects for projects to be developed individually or in small groups.

 

In consideration of the type of activity and teaching methods adopted, the frequency of this training activity requires the prior participation of all students in Modules 1 and 2 of safety training in the places of study, in e-learning mode.

Assessment methods

 

• Project to be presented orally at the end of the course or submitted successively as a written report.
• Oral exam about theory and assignments.

Teaching tools

e-learning platform: Virtuale

Office hours

See the website of Fabiana Zama