- Nick Hale - 'Rectangular pseudospectral differentiation matrices' or, 'Why it's not hip to be square'
Boundary conditions in pseudospectral collocation methods
are imposed by removing rows of the discretised differential operator and
replacing them with others to enforce the required conditions at the boundary.
A new approach, based upon projecting the discrete operator onto a lower-degree
subspace to create a rectangular matrix and applying the boundary condition
rows to ‘square it up’, is described.
We show how this new projection-based method maintains
characteristics and advantages of both traditional collocation and tau methods.
- Cameron Hall - 'Discrete-to-continuum
asymptotics of functions defined as sums'
When attempting to
homogenise a large number of dislocations, it becomes important to express the
stress in a body due to the combined effects of many dislocations. Assuming
linear elasticity, this can be written as a simple sum over all the
dislocations. In this talk, a method for obtaining an asymptotic approximation
to this sum by simple manipulations will be presented. This method can be
generalised to approximating one-dimensional functions defined as sums, and
work is ongoing to achieve the same results in higher dimensions.
- Vladimir Zubkov - 'On the tear film modeling'
A great number
of works about the tear film behaviour was published. The majority of these
works based on modelling with the use of the lubrication approximation. We
explore the relevance of the lubrication tear film model compare to the 2D
Navier-Stokes model. Our results show that the lubrication model qualitatively
describe the tear film evolution everywhere except region close to an eyelid
margin. We also present the tear film behaviour using Navier-Stokes model that
demonstrates that here is no mixing near the MCJ when the eyelids move relative
to the eyeball.
- Kostas Zygalakis - 'Numerical methods for stiff stochastic differential
equations'
Multiscale differential equations arise in the modelling
of many important problems in the
science and engineering. Numerical methods for such problems have been
extensively studied in the deterministic case. In this talk, we will discuss
numerical methods for (mean-square stable) stiff stochastic differential
equations. In particular we will discuss a generalization of explicit stabilized methods, known as Chebyshev methods to stochastic problems.
