MTH4340 - Numerical methods for partial differential equations

Overview

Partial differential equations are ubiquitous in many domains of sciences and industry, as they model phenomena with spatial and temporal variations. Most of these models are too complex to be exactly solved, and numerical methods are the only way to gather quantitative behaviour on the solutions. This unit covers the design, analysis and implementation of numerical methods for partial differential equations. Topics covered can include finite difference methods, finite element methods, finite volume methods, error analysis, elliptic equations, parabolic equations, implementation in dynamic languages (such as Python or Julia). The focus will be on the design of the methods, their mathematical analysis, and their implementation and numerical testing.

Offerings

S2-01-CLAYTON-ON-CAMPUS

Rules

Enrolment Rule

Contacts

Chief Examiner(s)

Professor Santiago Badia

Unit Coordinator(s)

Professor Santiago Badia

Notes

This level 4 unit and its level 3 counterpart MTH3340 share the same core content and learning activities such as seminars and applied classes. However, studies at Level 4 are distinguished from those at Level 3 by a deeper understanding of mathematical theories and their applications, higher levels of critical thinking, and greater autonomy in learning.

Learning outcomes

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

Critically evaluate and articulate the necessity of numerical methods for obtaining quantitative information on the solutions to partial differential equations;

Design, analyse, and rigorously assess the convergence and stability of numerical methods for a wide range of partial differential equations, demonstrating a deep understanding of the underlying mathematical principles;

Select and justify the use of advanced discretisation techniques based on their specific characteristics and the features of the considered mathematical model, showcasing expertise in tailoring methods to complex problems;

Implement advanced numerical methods in high-level programming languages such as Python or Julia, and critically interpret and analyse the resulting numerical outputs, demonstrating proficiency in computational skills;

Communicate complex theoretical and practical numerical problems involving partial differential equations with clarity and precision, both in written and oral forms, suitable for academic and professional audiences.

Teaching approach

Active learning

We will use a blend of lectures, applied classes and labs. Lectures will enable us, as usual in mathematics, to deliver the main content of the unit albeit in an interactive way, questioning your understanding and welcoming questions from you. Applied classes revolve around you actively solving exercises, putting in practice the content covered in the lectures; this is an essential way of assimilating that content. Labs will be focused around practical implementation of the methods, the issues around those and the interpretation of the numerical outputs.

Handbook