| 10:00 | Welcome |
| 10:05 | Roland Hunt (Strathclyde), Solution of steady fluid flow problems using Newton's iteration |
| 10:55 | Coffee/Tea |
| 11:25 | Phil Gresho (Lawrence Livermore National Laboratory, USA), Thermal convection in an 8:1 sidewall-heated enclosure |
| 12:15 | Lunch |
| 13:30 | Andrew Stuart (Warwick) Extracting macroscopic behaviour from dynamics |
| 14:20 | Gerald Moore (Imperial) Spectral methods for dynamical systems |
| 15:10 | Coffee/Tea |
| 15:40 | Marco Marletta (Leicester) Block operator matrices and the Ekman flow |
| 16:30 | Close |
Lawrence Livermore National Laboratory, USA. (pgresho@llnl.gov)
When Prof. Klaus-Jergen Bathe invited me to design a minisymposium to take place during his "1st M.I.T Conference on Computational Fluid and Solid Mechanics" (June, 2001), I was happy to respond with a 2-D model problem that was a spinoff from our challenging ongoing 3-D thermal convection simulations in support of the big laser program (NIF: National Ignition Facility) at LLNL (Lawrence Livermore National Laboratory). We simply selected a 2-D slice thru one of our 3-D geometries and chose the two problem parameters (Ra = 3.4X10^5, the Rayleigh number; and Pr = 0.71, the Prandtl number) relevant to our own work. Little did we know how prescient we were, as we only learned later --- from the person selected to perform the so-called Benchmark Solution via a Chebychev collocation spectral method, Dr Patrick Le Quere in France --- that the Rayleigh number selected was in a very small range (from ~3.2X10^5 to ~ 3.5X10^5) in which there exists but a SINGLE (periodic-in-time) solution. For near-by values of Ra, there are two, or even three, stable periodic solutions. Even at that, the 'history' of this CFD test problem became one of 'code-breaker', as many of the contributors required several trys (and some 'hints' from us) before they were able to 'push' their code toward the right 'solution space'. Related to this, it turned out that any method/algorithm that introduced ANY numerical dissipation (e.g., upwinding, streamline diffusion, backward Euler time integration) would find --- in most cases --- that their simulations went to a spurious steady state.
I will try to provide a useful summary of this interesting and rather challenging test problem as well as the results presented by the two dozen or so contributions that included 'all' of the current CFD technologies: FDM,FEM,FVM, and spectral. I will also describe the several contributions that focussed on the stability aspects of near-by solutions, such as the 'critical' Ra at which the steady flow first transitions to a time-varying one ( at Ra = ~ 3.1X10^5) --- leading to periodic behaviour with multiple cells ( ~ one dozen) and travelling waves because of a boundary layer instability close to the vertical walls.
Department of Mathematics, University of Strathclyde, Glasgow. (rh@maths.strath.ac.uk)
The numerical solution of the steady Navier-Stokes equations involves replacing the derivatives by differences and solving the ensuing nonlinear algebraic system. One way to solve this system is to use Newton's method, which, being second order convergent, is guaranteed to converge in typically 3 or 4 iterations, provided a suitable initial iterate is available. For this type of problem such an initial iterate is readily obtained, and a numerical solution can be obtained without recourse to upwinding, artificial viscosity or any conditionality constraint. Exploiting the banded nature of the Jacobian results in a reasonably efficent program.
We will illustrate the method by the benchmark problem of the fluid flow in a stepped-down channel, and the flow in a bifurcation in two and three dimensions. The latter is motivated by blood flow in an arterial bifurcation, which is a preferred site of atherosclerosis; a major cause of cardio-vascular disease. In three dimensions we will use a mix of traditional second order central and pseudospectral differences to produce an efficient, accurate code.
A general Newton package for solving solving any set of simulutaneous equations on a two-dimensional grid has been developed. If time permits, its use in solving free boundary problems, problems with integral constraints and implicit time-dependant problems will be discussed.
Department of Mathematics and Computer Science, University of Leicester. (mm7@mcs.le.ac.uk)
We consider block operator eigenvalue problems, which are eigenvalue problems of the form $Ax = \lambda Bx$ in which $A$ and $B$ are matrices whose elements are differential operators. Problems of this form arise in numerous applications, most notably in MHD. We shall give examples which show that the spectra of such problems can be very different from the spectra of differential equation problems, and describe necessary and sufficient conditions for a block operator problem to be reducible to a differential equation problem.
A particularly interesting example is the block operator problem describing the Ekman boundary layer. For this we locate the essential spectrum and develop a Titchmarsh-Weyl coefficient $M(\lambda)$. This allows a rigorous analysis of the regularizations of the Ekman problem, and in particular a proof of their spectral exactness.
If time permits, we shall give examples which show that regularizations of singular non-normal problems are often not spectrally exact, and describe a simple test which can be applied to sift out spurious eigenvalues.
This is joint work with Leon Greenberg of the University of Maryland.
Department of Mathematics, Imperial College, London. (g.moore@ic.ac.uk)
The standard discretisation method when analysing parameter- dependent autonomous systems has been collocation with piecewise polynomials: e.g. the packages AUTO and HOMCONT have used derivatives of COLSYS. We consider the advantages of using spectral approximation instead. In particular, Fourier spectral methods for periodic orbits and Laguerre spectral methods for homoclinic orbits and periodic connections are discussed.
Mathematics Institute, University of Warwick. (stuart@maths.warwick.ac.uk)
Recent work of Dellnitz, Deuflhard, Schutte and others has shown that transfer operators can be effectively used to extract low-dimensional dynamical information from deterministic and random dynamical systems. In this talk I will describe an attempt to shed light on this methodology through the study of simple model problems which have the property of exhibiting low dimensional stochastic behaviour in macroscopic variables. The work is in collaboration with W Huisinga (FU Berlin), C Schutte (FU Berlin), J. Terry (Warwick) and P. Tupper (Stanford).