**Prerequisites**

- ODEs: solution of systems of linear ODES (diagonalization, variation of coefficients).
- PDEs: Classification of second order PDEs (hyperbolic, parabolic, elliptic PDEs).
- Numerical solution techniques: Euler method for ODEs, Fixed point methods, Newton method, interpolation, least squares, midpoint and trapezium integration rule.
- Analysis: Taylor series, Gauss' divergence theorem, Jacobian matrix
- Linear Algebra: Normal matrices, diagonalization, Jordan normal form, Schur normal form.
- MATLAB: basic ability in programming in MATLAB

**Aim**

The aim is to make students familiar with the basic building blocks in the design of numerical methods for large-scale bifurcation problems for both steady-states and periodic solutions.

**Description**

Large-scale systems arise in many different fields, such as computational fluid dynamics, ocean and climate models, chemical engineering, simulation of large electronic circuits, etc. Sometimes these models are large systems of Ordinary Differential Equations (ODEs) or Differential Algebraic Equations (DAEs), but often the models are described by Partial Differential Equations (PDEs). However, for a numerical bifurcation analysis, the latter are often space-discretized and then treated as a large ODE or DAE system.

If these equations are nonlinear then multiple solutions may exist for certain values of the parameters in the model. The aim of bifurcation analysis is to make on overview of all solutions in a certain range of the parameters and moreover to determine whether such a solution is stable or not. This approach gives fundamental insight in the behaviour of the system. Moreover, if we are able to solve the PDEs to a sufficient high degree of accuracy, we can quantitatively predict the behaviour of the phenomenon it is describing.

The goal is to make students familiar with the basic building blocks in the design of numerical methods for large-scale bifurcation problems for both steady-states and periodic solutions. During the course, computer exercises will have to be made in order to get familiar with the numerical behavior of the methods.

Topics to be covered in the course:

- space- and time discretizations,
- solution of nonlinear problems,
- classical methods for eigenvalue problems with application to stability analysis,
- Krylov subspace methods for large sparse eigenvalue problems (Arnoldi) and linear systems (GMRES, BiCGstab etc.),
- continuation and stability analysis of steady states,
- continuation and stability analysis of periodic solution with (i) methods that extend the small systems approach (ii) the time-simulation based approach,
- review of existing methods and software,
- some applications.

**Organization**

Each class consists of three 45 minutes slots of lecture.

Each week a homework exercise will be handed out which will often contain a programming part. At the end of the course there will be a labsession where computation will be done with an advanced ocean model.

**Lecturers**

F.W. Wubs

Johann Bernoulli Institute, University of Groningen,

P.O.Box 407, 9700 AK Groningen, The Netherlands

e-mail: f.w.wubs@rug.nl - Web page: http://www.math.rug.nl/~wubs

H.A. Dijkstra

Institute for Marine and Atmospheric Research Utrecht, Utrecht University,

Princetonplein 5, 3584 CC Utrecht, The Netherlands

e-mail: dijkstra@phys.uu.nl - Web page: http://www.phys.uu.nl/~dijkstra

- Docent: Hendrik Dijkstra
- Docent: Fred Wubs