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DOC: pressure-correction scheme for unsteady Navier–Stokes #240

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gdmcbain opened this issue Nov 8, 2019 · 1 comment
Open

DOC: pressure-correction scheme for unsteady Navier–Stokes #240

gdmcbain opened this issue Nov 8, 2019 · 1 comment

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@gdmcbain
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@gdmcbain gdmcbain commented Nov 8, 2019

Following the Newton steady solution of the Navier–Stokes equation in ex27 and the one-dimensional linear stability analysis in ex29, the next step is to show how to implement an algorithm for the unsteady Navier–Stokes equations.

While this could be done with a fairly straightforward extension of the steady solution (e.g. backward Euler or higher backward differentiation formula), a popular approach is the class of approximate ‘projection methods’ which decouple pressure and velocity. This is what's used in the FEniCS tutorial. So far I've only translated tutorial 07 into scikit-fem in gdmcbain/fenics-tuto-in-skfem#4. It's not very interesting so I propose to develop something else for here but till then that serves as a bit of an example.

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@gdmcbain gdmcbain commented Nov 8, 2019

One thing that I noticed is that whereas the quadrature rules defined in the velocity and pressure interior bases in the previous examples of the Stokes and Navier–Stokes equations, ex18, ex24, ex27, e.g.

basis = {variable: InteriorBasis(mesh, e, intorder=3)
for variable, e in element.items()}

have the intorder fixed to a common value which allows mixed bilinear forms (e.g. for pressure-gradient term in momentum equation or the velocity-divergence term in the continuity equation) to be assembled, the order isn't high enough when the velocity-mass bilinear form is required, as it is in most unsteady schemes. The fix is simply to increase the common intorder to 4. This was already done silently for the same reason in the linear hydrodynamic stability example ex29 #237.

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