Adaptive timestepping for incompressible flow problems
Simulation of the motion of an incompressible fluid remains an important but very challenging problem. The resources required for accurate three-dimensional simulation of practical flows test even the most advanced computer hardware. The necessity for reliable and efficient solvers is widely recognised. Two fundamental components of such a solver are: the error control used for self-adaptive time stepping; and the linear solver used at each time level. This talk will focus on the first component.
Conventional incompressible flow codes typically use semi-implicit time integration leading to a Poisson or Stokes-type system at every time step, but with a stability restriction on the time step. Our alternative approach is a stable version of the so-called TR-AB2 "smart integrator" developed by Gresho and Hindmarsh in the 1980's. Such fully-implicit methods have no restriction on the time step, but have only become feasible in recent years because of developments in solution techniques for the linear (Oseen) systems that arise at each time level. A theme that will emerge in the talk is that even simple convection-diffusion problems have complex time-scales---so that an adaptive integrator is essential if accurate solutions are to be computed efficiently.
This is a joint research project with Phil Gresho (ex-LLNL, California) and David Griffiths (University of Dundee). The impetus for the work was an EPSRC Visiting Fellowship grant GR/R26092 funded by the Computational Engineering Mathematics initiative.