# SYMPLECTIC NUMERICAL METHODS - 2022/3

Module code: MATM051

## Module Overview

Structure preserving discretizations are an important tool for the numerical computation of the long-time behaviour of mechanical systems and are widely used in molecular dynamics simulations, fluid dynamics and climate modelling. In this module we will cover the fundamental ideas of structure preserving numerical methods concentrating on symplectic integrators.

### Module provider

Mathematics

DIAZ TORRES Alexis (Physics)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 84

Lecture Hours: 33

Captured Content: 33

Semester 2

None

## Module content

Introduction: Introduce examples of Hamiltonian systems from physics and their conserved quantities; motivate symplectic integrators versus non-symplectic integrators for some simple examples, in particular with respect to numerical conservation of energy.

One-step methods for ODEs:  Introduce consistency and order of convergence for one-step methods; treat some low order Runge-Kutta methods as examples.

Hamiltonian systems and symplectic discretizations:  Define symplectic maps, show that Hamiltonian flows are symplectic. Introduce implicit midpoint rule as symplectic integrator. Introduce simple partitioned symplectic Runge-Kutta methods (symplectic Euler method) and symplectic discretizations constructed by splitting methods; apply to problems from physics.

Backward error analysis for symplectic integrators: Introduce the concept of a modified equation; derive exponential error estimates and approximate energy conservation for exponentially long times. Recall Noether's theorem and prove conservation of angular momentum of symplectic methods.

## Assessment pattern

Assessment type Unit of assessment Weighting
School-timetabled exam/test IN-SEMESTER TEST (50 MINS) 20
Examination EXAMINATION - WRITTEN EXAMINATION 80

N/A

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity to demonstrate:

• Understanding of and ability to interpret and manipulate mathematical statements.

• Subject knowledge through the recall of key definitions, theorems and their proofs.

• Analytical ability through the solution of unseen problems in the in-semester test and in the exam.

Thus, the summative assessment for this module consists of:

• One two hour examination (three of four best answers contribute to exam mark) at the end of the Semester; worth 80% module mark.

• One in-semester test; worth 20% module mark.

Formative assessment and feedback

Students receive written feedback via two unassessed, and the two assessed, coursework assignments over an 11 week period.  In addition, verbal feedback is provided by lecturer in lectures and in office hours.

## Module aims

• This module aims to introduce students to the concept of structure preserving numerical integration using the example of symplectic integrators for mechanical systems

## Learning outcomes

 Attributes Developed 001 Design and implement simple symplectic Runge-Kutta methods and splitting methods KCPT 002 Explain the advantages of symplectic discretizations vs non-symplectic integrators using backward error analysis KCP

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to introduce the students to the theory of structure-preserving discretizations.

The learning and teaching methods include:

• 3 hour lectures per week x 11 weeks, on the blackboard and Q + A opportunities for students;

• Including (every second week) a tutorial lecture for guided discussion of solutions to problem sheets or unassessed coursework provided to and worked on by students in advance.

• revision lectures (in week 12).

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.