# Structure-preserving discretization of differential equations

### Atlantic Association for Research in the Mathematical Sciences - SUMMER SCHOOL

### Dalhousie University, Halifax, Nova Scotia (Canada)

Final exam |
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Exam problems (part 2) |

Exam problems part 1. |

Final exam part 2 with answers |

Messages |
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Solution to Assignment 3 assignment3_with_answers Script commfree4.m cfree4step.m (needed by commfree4.m) |

At these links (link1 link2 ) you can find information regarding funding opportunities if you want to visit Norway for an exchange during your studies. |

An updated note on Lie groups is now available under Lecture 10 and Lecture 11. |

The third assignment is now published under assignments. |

Dear all, the deadline for handing in the second assigmnet is moved 24 hours forward. New deadline Thursday 23rd at 15:00. |

This note might be useful in order to get things right in assignment 2:note on assignment 2. |

The lecture notes for Lecture 5 on Variational integrators are now available |

Thanks to all the students for having done a real nice job on the first assigment. All of you pass either with A or A+. |

To students working with the 1st assignment on Structure preserving discretization of differential equations, if you need more time you can have an extra 24 hours to finish. We'll be in the computer lab from 13:30 today for supervisions. |

New notes on lecture 3 and 4 are now available. |

Printing in the lab: I've been told that the IT services require all students in our class to submit their netID's and passwords in order to be able to print in the lab. Then Queena will set it up. Please send your netID and password via e-mail to Queena. |

The first assignment is now published under Assignments |

Friday our class will be in the computer lab (room 301 in DUNN building). Thodore will help us get in the room on Friday at 3:30 pm. All students should have received a guest account when they registered for the residence. If you do not have your own computer, with Matlab (or other similar software tools) installend, then you can use the computers at the computer lab logging in with your guest account. Please, make sure to have your guest account info at hand on Friday. |

First lecture: 6th of July 2015 in DUNN building room 117, from 15:30. |

## Lecturers

## Lectures

Mondays to Fridays 15:30-17

## Description and learning outcome

This course addresses the structure preserving numerical integration of differential equations with an underling geometric structure. We focus on problems whose flows preserve first integrals, have an underlying symmetry, are completely integrable, are symplectic etc. We address the design and study of numerical methods which preserve the underlying geometric structure under numerical discretization, the so called geometric numerical integrators. The students will learn about a range of problems where geometric numerical methods are a natural choice and discuss where and when these integrators constitute a real advantage with respect to more classical general purpose integrators (e.g. classical Runge-Kutta methods and multi-step methods). The practical implementation in Matlab of some of the numerical techniques will be pursued as part of the learning activities, as well as the design and execution of appropriate numerical experiments aimed at testing the features of the methods in a selection of simple but relevant problems. We will focus in particular on three main topics:

- Symplectic integration of Hamiltonian systems including variational integrators;
- Lie group integrators, coordinate free numerics, geometric numerical integration on manifolds;
- Preservation of invariants (energy) and of modified invariants.

### On successful completion of the course, students will be able to

- Explain why structure preserving methods are needed, which purpose they serve, and to distinguish cases in which such integrators may be preferred over other integrators
- Identify the main classes of structure preserving differential equations and be able to select numerical integrators that will preserve the various structures exactly or approximately
- Analyse the effect of using particular structure preserving integrators to the respective problem types
- Implement schemes for small test problems in order to identify through experiments the benefits and challenges with structure preserving integrators