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dajuno May 5, 2017 19:51

Fractional step-Theta time stepping and SUPG stabilized FEM
Dear all,

I am working on a FEM based incompressible Navier-Stokes solver.
I have implemented the second order Fractional step-Theta time discretization (see [1], [2]; only time-stepping, NO splitting! has slightly better stability properties than Crank-Nicolson) for inf-sup stable elements (P2P1), which corresponds roughly to staggered grids in FVM.

I also use an unstable P1/P1 discretization, stabilized with SUPG/PSPG or Galerkin/Least-squares (GLS), as in [3], [4], that I want to apply the FS-Theta method to.
However, I wonder how to bring both parts together: How to use the Fractional step-Theta discretization on the GLS stabilization terms.
The problem is that different parts of the FS-Theta discretized Navier-Stokes equations are multiplied by different constants (thetas), i.e., du/dt, convection + diffusion term, and the pressure gradient, respectively.
It seems logical to add weightings correspondingly to the GLS formulation, but I am really not sure (what and where) and I have not found any references for that.

I realize that this is a very particular question... any advice would be very much appreciated!!

[1] Turek 1999, Efficient solvers for incompressible flow problems: an algorithmic and computational approach (Book)
[2] John 2006, A comparison of time-discretization/linearization approaches
for the incompressible Navier–Stokes equations
[3] Tezduyar 1992, Stabilized finite element formulations for incompressible flow computations
[4] Shakib & Hughes, 1991, Part X

Kind regards

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