- 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.