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.

Areas of study: Applied mathematics Pure mathematics Mathematics Mathematical statistics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

The Handbook lists no prerequisite, corequisite or prohibition for this unit.

Learning outcomes

1. Critically evaluate and articulate the necessity of numerical methods for obtaining quantitative information on the solutions to partial differential equations;
2. 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;
3. 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;
4. 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;
5. 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.

Workload

• Three 1-hour seminars; • One 2-hour applied class (in weeks 2-12) and • 7 hours of independent study per week

Student experience, not official rules. Nothing here changes what the Handbook says.