# Mathematics

## MTH2005 - Modelling: Theory and Practice (2020)

MODULE TITLE CREDIT VALUE Modelling: Theory and Practice 30 MTH2005 Prof Bob Beare (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 11 0
 Number of Students Taking Module (anticipated) 128
DESCRIPTION - summary of the module content

The role of the mathematician has changed significantly with the advent and increasing processing power of modern computers. This module develops the theoretical and practical skills necessary to develop and apply numerical methods of solution, using a computer package such as Matlab. Drawing on techniques learned in the first year, in this module you will analyse the underpinning mathematics and the practical coding challenges appropriate to applying them compute solutions to problems in a variety of situations. For part of the module you will work together within groups to develop and analyse your own computer models. You will study the performance of underlying algorithms and the limits to their predictive power. Specifically, you will study approximation methods for root finding, optimization, integration and solution of differential equations. The skills developed here will be useful for modules such as MTH3039 Computational Nonlinear Dynamics and more generally for coding and simulation in any application area of scientific computing.

Prerequisite modules: MTH1003 or NSC1002 (Natural Science Students) or equivalent.

Corequisite module: MTH2003.

AIMS - intentions of the module

This module explores the use of computers to solve mathematical problems by means of numerical approximation. The techniques discussed form the basis of the numerical simulation and computer modelling of problems in science and business. The key aim is developing an understanding of the numerical algorithms and we will explore these both theoretically and through case studies that develop further the mathematical modelling techniques learned in MTH1003.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module, you should be able to:

Module Specific Skills and Knowledge:

1 demonstrate a working knowledge of the theory and practical implementation of basic numerical methods;

2 explore applications and ideas underpinning more advanced methods that are developed in third/fourth stage modules and project work;

3 develop and code your own mathematical models with guidance;

4 interpret the outputs from your models, drawing suitable conclusions from your data;

5 evaluate the effectiveness of your models at explaining and predicting the phenomena you are modelling.

Discipline Specific Skills and Knowledge:

6 explore the subject material of the module through diverse applications to areas of science and business;

7 use computation as a natural method for tackling such problems;

Personal and Key Skills

8 demonstrate theoretical and practical mathematical skills, including programming.

9 formulate and solve problems independently;

10 communicate computer results and mathematical derivations effectively.

11 work in teams and use a variety of sources to produce reports and other appropriate scientific outputs.

SYLLABUS PLAN - summary of the structure and academic content of the module

Root Finding

Bisection, Newton-Raphson and fixed point convergence. Proofs of convergence and non-convergence. Demonstration of convergence and non-convergence using diagrams.

Quadrature and Ordinary Differential Equations (ODEs)

Finite differences, including first and second-order approximations for both the first and second derivative. Timestepping of a first-order ODE using the following methods: forward Euler, leap-frog, Runga-Kutta, Implicit, Adams-Bashforth and Adams-Moulton. Understanding of numerical stability, including identifying the true solution.  Analysis of both accuracy and stability of timestepping methods.

Matrices

The LU decomposition and Gaussian elimination for matrix inversion. Iterative matrix inversion methods: Jacobi, Gauss-Seidel and Successive Over Relaxation. Analysis of convergence of the iterative methods using: (1) method of norms and (2) Spectral radius. The condition number. Calculating eigenvalues using the power method.

Partial Differential Equations (PDEs)

1d diffusion and 1d advection equation as prototypical PDEs. Finite difference methods. Implicitness and possible extensions e.g. semi-lagrangian and montone advection.

Case studies will be developed in a variety of areas of mathematics, science and/or business as part of the coursework, to support and enhance the material formally presented in lectures/tutorials.  These case studies might include, for example: optimisation; Fast Fourier Transform; stochastics; modelling of data.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 64 236
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 44 Lectures Scheduled learning and teaching activities 10 Practicals in a computer lab Scheduled learning and teaching activities 10 Tutorials Guided independent study 236 Lecture and assessment preparation; wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Exercise sheets 5 x 5 hours All Discussion in tutorials; model solutions where appropriate.

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 70 30
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 30 2 hours   Via SRS
Case studies 2 x 20 5000 words or equivalent   Written comments on returned coursework, customized marksheet
Project 30 7500 words or equivalent   Written comments on returned coursework, customized marksheet

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-reassessment
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.

If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark.

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE – http://vle.exeter.ac.uk