Math Courses

Advanced topics in numerical analysis (2/3)

by Francesco Tudisco (GSSI), Nicola Guglielmi (GSSI)

→ Europe/Rome
Description

Lecturers
Nicola Guglielmi, nicola.guglielmi@gssi.it
Francesco Tudisco, francesco.tudisco@gssi.it

Course description
This course is an introduction to modern numerical analysis. The primary objective of the course is to develop graduate-level understanding of computational mathematics and skills to solve a range real-world mathematical problems on a computer by implementing numerical
algorithms with MATLAB.

Course requirements
Calculus and basic numerical analysis. Previous programming experience in any language may help

Examination and grading
Written exam and practical (computational) assessment

Course content
The course will cover the following topics

  • Numerical quadrature
    – Order conditions
    – Error analysis
    – Superconvergence
    – Orthogonal polynomials
    – Gaussian quadrature
     
  • Linear multistep methods for ODEs
    – Explicit and implicit Adams’ methods
    – Local error and stability
    – Convergence
    – Variable step size multistep methods
    – General linear multistep methods
     
  • Runge Kutta methods for ODEs
    – General form
    – Convergence theory
    – Order conditions
    – Stability theory
    – A stability
    – B stability
    – Stiff problems
    – Von Neumann theorem
    – Application to evolution PDEs
     
  • Iterative methods for sparse linear systems
    – General projection methods
    – CG and GMRES
    – Preconditioning strategies
    – Multigrid methods
     
  •  Boundary value problems (BVP)
  • Rayleigh Ritz Galerkin methods
  • Finite elements for BVP


Books of reference
E. Hairer, G. Wanner, S. P. Nørsett; Solving Ordinary Differential Equations I
E. Hairer, G. Wanner; Solving Ordinary Differential Equations II
Y. Saad; Iterative methods for Sparse Linear Systems (Free Online Version)