David Silvester 
universite de Manchester,

Title:
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A Review of Solution Techniques for Unsteady Incompressible Flow


Abstract:
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The necessity for reliable and efficient incompressible flow solvers   is
widely recognised. This talk will focus on two components of such a  
solver: the error control used for self-adaptive time stepping; and   the
linear solver used at each time level.
Conventional 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 TR-AB2 smart integrator originally  
developed by Gresho in the 1980's. Such fully-implicit time
integration methods have no restriction on the time step, but have   only
become feasible in the last five years because of developments in  
solution techniques for the linear (or linearized) Oseen systems that  
arise at each time level. To this end, the preconditioning framework  
that we propose offers the possibility of fast convergence that is  
robust with respect to the problem parameters (namely; the mesh size,  
the time step and the Reynolds number).
The preconditioning framework that is proposed can be readily extended  
to multi-physics models that include the Navier-Stokes equations as a  
component. We illustrate this by extending our methodology to the case  
of the Boussinesq model of bouyancy driven flow.