Aoussou, J.P., 2016. An Iterative Pressure-Correction Method for the Unsteady
Incompressible Navier-Stokes Equation. SM Thesis, Massachusetts Institute of Technology, Computation for Design and Optimization Graduate Program, June 2016.
The pressure-correction projection method for the incompressible Navier-Stokes equation
is approached as a preconditioned Richardson iterative method for the pressure-
Schur complement equation. Typical pressure correction methods perform only one
iteration and suffer from a splitting error that results in a spurious numerical boundary
layer, and a limited order of convergence in time. We investigate the benefit of
performing more than one iteration.
We show that that not only performing more iterations attenuates the effects
of the splitting error, but also that it can be more computationally efficient than
reducing the time step, for the same level of accuracy. We also devise a stopping
criterion that helps achieve a desired order of temporal convergence, and implement
our method with multi-stage and multi-step time integration schemes. In order to
further reduce the computational cost of our iterative method, we combine it with an
Aitken acceleration scheme.
Our theoretical results are validated and illustrated by numerical test cases for
the Stokes and Navier-Stokes equations, using Implicit-Explicit Backwards Difference
Formula and Runge-Kutta time integration solvers. The test cases comprises a now
classical manufactured solution in the projection method literature and a modified
version of a more recently proposed manufactured solution.