Solving parabolic problems without time-stepping
Ian Sloan
Department of Mathematics, University of New South Wales
I.sloan@unsw.edu.au
This joint work with Vidar Thomée (Chalmers) and Dongwoo Sheen (Seoul) adopts a different approach to certain parabolic problems: instead of carrying the solution forward through small time increments, we make use of a contour integral representation of the Laplace transform of the solution. After a suitable contour deformation, the solution is approximated by a special quadrature formula. To each quadrature point there corresponds a (complex-valued) elliptic problem, which can be solved approximately in one of the standard ways.
The first phase of the work (Math. Comp., 2000) considered only homogeneous problems. In current work we are now tackling inhomogeneous problems, by suitably approximating the inhomogeneous terms.
The approach is inherently parallelisable (in contrast to step-by-step methods), because the finite set of elliptic problems can be solved in any order. The method seems to be of possible practical interest for appropriate problems.
The numerical solution of free surface flows
Sean McKee
Department of Mathematics, University of Strathclyde
s.mckee@strath.ac.uk
This talk will describe a Marker and Cell (MAC) type code developed over the last decade with a four-person Brazilian team. It was first motivated while the speaker was at Unilever in connection with container filling of low viscosity Newtonian and non-Newtonian fluids.
The simplified marker and cell algorithm with many modifications and extensions will be described as will the solid modelling code in which it is embedded. Examples of container filling flows will be presented and experimental work undertaken at Unilever Research will be described. Jet buckling is computed and a tentative criterion for buckling will be proposed. Axisymmetric flows will be simulated and a comparison will be made with GI Taylor's jet impingement experiments. Drop splashing will be displayed and a hydraulic jump will be simulated-the latter being used as a realistic test problem for a comparison of some of the recent high order upwinding schemes. Future and ongoing work will be discussed.
Computation of connections to periodics
Gabriel Lord
Department of Mathematics, Heriot-Watt University
gabriel@ma.hw.ac.uk
We will discuss the computation of homoclinic connections to periodic orbits in reversible and non-reversible systems. We consider a model example (non-reversible) and compute the connection from a fixed point to an unstable periodic orbit. As a main example we take a problem arising from water waves and examine numerically homoclinic connections to periodics and in particular the small amplitude limit. Finally we present some preliminary results from a model system of dendritic tissue on heteroclinic connections between periodic solutions.
Geometric integrators and molecular dynamics applications
Ben Leimkuhler
Department of Mathematics, Leicester University
B.Leimkuhler@mcs.le.ac.uk
Molecular N-body problems present many issues for the simulator, including challenges due to large dimensionality, nonlinear complexity, and the need to obtain solutions on lengthy time intervals. Typically, the trajectories themselves are not inherently interesting, but provide a means of computing an approximate average of some physical quantity with respect to an appropriate density of states (e.g. in the microcanonical, canonical or constant temperature and pressure ensemble).
Geometric integrators are timestepping methods that preserve invariants or symmetries associated to the flow-map of a dynamical system. The most important instances of geometric integrators include unitary integrators (e.g. quantum propagators), symplectic integrators for Hamiltonian systems, and time-reversible integrators. Geometric integrators have become quite popular in recent years, particularly as tools for long term physical or chemical simulation. There is growing evidence---both empirical and analytic---to justify the use of these schemes.
In this talk, I will discuss some of the algorithmic issues arising in molecular dynamics simulation, including the treatment of rigid body systems and a generalized Feynmann path-integral quantum-molecular dynamics method, and show how some of the problems can be tackled with efficient geometric schemes.
Discontinuous hp-finite element methods for advection-diffusion problems
Endre Süli
Computing Laboratory, University of Oxford
Endre.Suli@comlab.ox.ac.uk
We consider the hp-version of the discontinuous Galerkin finite element method for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, first-order hyperbolic equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilisation. In the hyperbolic case, an hp-optimal error bound is derived. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by half a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For element-wise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.
Nonlinear programming and free boundary problems
Roger Fletcher
Department of Mathematics, University of Dundee
fletcher@maths.dundee.ac.uk
Recent developments in software for large scale nonlinear programming (NLP), modelling languages such as AMPL that provide automatic differentiation, and internet access via the NEOS system, make it very easy for the non-expert user to apply NLP techniques. The new opportunities that present themselves are illustrated with a number of case studies in PDEs, including obstacle problems, packaging problems, melting (Stefan) problems, and possibly some others. Some other issues are also considered including the modelling of complementarity constraints and the use of conditional constructions.