Numerical analysis is the study of algorithms for the problems of continuous mathematics, that is, problems involving real or complex variables.
For such problems exact, closed-form solutions usually do not exist or are not readily obtainable, so numerical analysts develop and analyse algorithms that compute numerical approximations.
The Manchester group works across the spectrum from theory to algorithms to developing open source software. Our research focuses particularly on
- Finite Element Approximation
- Numerical Algorithms and Software Development
- Numerical Linear Algebra
- Numerical Linear Algebra and Solvers for PDEs
- Optimization and Compressive Sensing
- Stochastic Differential Equations
- Uncertainty Quantification
The Numerical Analysis Group currently has over 30 members, comprising staff, research associates, PhD students and visitors, and is very active in organising conferences and workshops, publishing papers and supervising PhD students (see some recent theses).
We are interested in the numerical simulation of physical processes; such as fluid flow, heat and mass transfer, and large deformation of solid bodies, and particularly the interaction between such processes. Typical research projects start with the development of suitable mathematical models for the physical system; this is followed by the formulation of appropriate discretisations for the governing equations and the design of efficient methods for their numerical solution on state-of-the-art computers. Computational studies are often complemented by asymptotic analyses, and many projects involve collaborations with experimentalists (particularly at the MCND). The development and optimisation of numerical methods frequently involves close interactions with members of the Numerical Analysis group, particularly the Finite Element Group.
Current work includes the development of an object-oriented finite element library (oomph-lib) for the simulation of nonlinear multi-physics problems, such as large-displacement fluid-structure interaction problems.
As a representative example, the figure below shows some recent results from the numerical simulation of 'pulmonary airway closure', a surface-tension-driven, fluid-elastic instability of the lung's liquid lining that leads to the occlusion of the airway with a liquid bridge. The figures show different stages of this instability:
Initially, a thin liquid film lines the axisymmetric airway wall (top left), a primary axisymmetric instability causes the redistribution of fluid along the airway (top right), surface-tension creates a strong compressive load on the airway wall in the region where the film thickness increases. When this compression becomes sufficiently large, the airway buckles non-axisymmetrically (bottom left), and ultimately becomes completely occluded by the fluid (bottom right). The results were obtained from a finite-element-based solution of the unsteady, 3D Navier-Stokes equations (describing the fluid flow), coupled to the equations of large-displacement shell theory (describing the deformation of the elastic airway wall).