Partial differential equations are a central modelling tool in applied mathematics and mathematical physics. They also play an important role in pure mathematics, not least as a stimulus in the development of concepts and methods of classical and modern analysis. This module provides an introduction to the classical analytical treatment of second-order linear partial differential equations and techniques for their numerical solution. The essential concepts and methods are introduced and developed for prototype partial differential equations representing the three classes: parabolic; elliptic; hyperbolic. Finite difference and finite element approximations to the solutions of partial differential equations are developed. The accuracy and stability of the numerical schemes are investigated. Direct and iterative methods for solving the linear systems arising from the numerical approximation of partial differential equations are described.

Prerequisite Modules: MA0122 Analysis I, MA0126 Analysis II

Recommended Modules: MA0212 Linear Algebra, MA0221 Analysis III, MA0232 Modelling with Differential Equations

• Use the method of characteristics to solve elementary first order partial differential equations.
• Classify second-order partial differential equations.
• Interpret the concept of well-posedness of initial/boundary value problems.
• Recognise the characteristic properties of basic partial differential equations and their solutions.
• Discretize partial differential equations using finite differences and finite elements.
• Analyse the convergence and stability properties of numerical approximations.
• Compare the relative merits of direct and iterative methods for solving the linear systems of equations arising from a discretization

54 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 100 hours private study including preparation of solutions for given exercises.

• Achieve precision and clarity of exposition.
• Recognise, formulate and solve problems in an interdisciplinary environment
• Appreciate the role of their skills in the interplay of sciences.
• Appreciate of the applicability of mathematical modelling techniques to a range of physical situations.
• Gain an awareness of important issues in the solution of differential equations using numerical techniques.

Formative assessment is carried out by means of regular exercises.  Feedback to students on their solutions and their progress towards learning outcomes is provided during lectures.  

The in-course element of summative assessment is an exercise similar in form to the tutorial exercises. This allows students to demonstrate a level of knowledge and skills appropriate to that stage in the module.

The major component of summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has a choice of four from six equally weighted questions.

• Methods for elementary first order equations.

• The heat equation
  • physical background, fundamental solution, solution of initial-value problems by convolution, uniqueness problems, maximum-minimum principles for initial-value problems.

• Numerical approximation of parabolic equations in one space dimension
  • finite difference approximations, truncation error, explicit and implicit schemes, convergence, discrete maximum principle, Crank-Nicolson scheme, Fourier error analysis.

• The potential equation
  • fundamental solution, Green’s function, maximum principles, properties of harmonic functions, Newton potentials.

• Numerical approximation of elliptic equations
  • weak formulation and the finite element method, triangular and quadrilateral elements, stiffness and mass matrices, error estimates, direct and iterative methods for solving linear systems.

• The wave equation
  • physical background, one-dimensional wave equation, solution and properties of the higher dimensional equation, energy integral method.

• Numerical approximation of hyperbolic equations in one space dimension
  • finite difference approximations, stability and the CFL condition, upwind scheme, Fourier error analysis, Lax-Wendroff scheme.

Partial Differential Equations, Evans, L. C., American Mathematical Society

Numerical Solution of Partial Differential Equations, Morton, K. W.,  & Mayers, D. F., Cambridge University Press

Partial Differential Equations, Copson, E. T., Cambridge University Press

Finite Elements and Fast Iterative Solvers, Elman, H., Silvester, D. J., & Wathen, A. J., Oxford University Press

Numerical Solution of Partial Differential Equations: Finite Difference Methods, Smith,G. D., Oxford University Press