Undergraduate Programme and Module Handbook 2017-2018 (archived)

Module MATH3081: NUMERICAL DIFFERENTIAL EQUATIONS III

Department: Mathematical Sciences

MATH3081: NUMERICAL DIFFERENTIAL EQUATIONS III

Type	Open	Level	3	Credits	20	Availability	Available in 2018/19 and alternate years thereafter	Module Cap		Location	Durham

Prerequisites

Numerical Analysis II (MATH2051) AND Analysis in Many Variables II (MATH2031).

Corequisites

One 20 credit Level 3 mathematics module.

Excluded Combination of Modules

Numerical Differential Equations IV (MATH4221)

Aims

To develop a basic understanding of the theory and methods of numerical solution for differential equations, both ordinary and partial.

Content

Introduction to numerical methods for initial-value problems.
Local and global truncation errors, convergence.
One-step methods, with emphasis on explicit Runge-Kutta methods.
Linear multistep methods, in particular the Adams methods
Predictor-corrector methods and estimation of local truncation error.
Stability concepts and stiff problems.
Finite difference methods for parabolic PDEs, CFL-condition, stability.
Finite element or finite difference method for elliptic equations.
Iterative methods to solve linear systems.
Eigenvalue problem for elliptic operators.
Finite volume or finite difference method for the wave equation, CFL condition.
Specialist software will be used for computational work throughout the module.

Learning Outcomes

Subject-specific Knowledge:

By the end of the module students will: be able to solve novel and/or complex problems in Numerical Differential Equations.
have a systematic and coherent understanding of theoretical mathematics in the field of Numerical Differential Equations.
have acquired a coherent body of knowledge of these subjects demonstrated through one or more of the following topic areas: Consistency, convergence and linear stability of numerical methods.
One-step methods, particularly Runge-Kutta methods.
Linear multi-step methods.
Predictor-corrector methods
Finite difference methods for parabolic and elliptic PDEs
Iterative methods to solve linear systems.
Eigenvalue problem for elliptic operators.

Subject-specific Skills:

In addition students will have specialised mathematical skills in the following areas which can be used with minimal guidance: Modelling, Computation.

Key Skills:

Modes of Teaching, Learning and Assessment and how these contribute to the learning outcomes of the module

Lectures demonstrate what is required to be learned and the application of the theory to practical examples.
Assignments for self-study develop problem-solving skills and enable students to test and develop their knowledge and understanding.
Formatively assessed assignments provide practice in the application of logic and high level of rigour as well as feedback for the students and the lecturer on students' progress.
The end-of-year examination assesses the knowledge acquired and the ability to solve predictable and unpredictable problems.

Teaching Methods and Learning Hours

Activity	Number	Frequency	Duration	Total/Hours
Lectures	40	2 per week for 19 weeks and 2 in term 3	1 Hour	40
Problems Classes	8	Four in each of terms 1 and 2	1 Hour	8
Preparation and Reading				152
Total				200

Summative Assessment

Component: Examination		Component Weighting: 100%
Element	Length / duration	Element Weighting	Resit Opportunity
Written examination	3 hours	100%

Formative Assessment:

Four written assignments to be assessed and returned in each of Michaelmas and Epiphany terms

■ Attendance at all activities marked with this symbol will be monitored. Students who fail to attend these activities, or to complete the summative or formative assessment specified above, will be subject to the procedures defined in the University's General Regulation V, and may be required to leave the University