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Old   August 22, 2014, 05:49
Default higher order discretisation scheme
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Simon Hahn
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Hello people,

A part of my final university thesis in computational fluid dynamics is to simulate the Navier-Stokes transport equation. After discretising it by applying 'finite volume method' I was on search for discretisation schemes.
The problem is, that I only have convective flux and no diffusive flux:

flux_east*phi_east - flux_west*phi_west = 0 (at first step steady-state without transient effects)

This leads to oszillations in the calculation of my general transported variable 'phi' by using 'QUICK' and even 'Hayase-QUICK' scheme. The reason is, that the coefficients (a_west, a_east) of the tri-diagonal matrix, which should remain negative, become positive in absence of the diffusive flux.

For my first approach I already used '1st order upwind' which was quite stable but the appearing numerical diffusion let me think about a higher order differencing scheme.

I've already worked with the book 'An Introduction to Computational Fluid Dynamics' by Versteeg and Malalasekra and also numerous researches on the internet couldn't give me an answer to this concrete problem.

So I wanted to ask, if anyone has already made the same experience and can suggest me a solution - could be just a hint, literature or an explicit discretisation scheme.
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Old   August 22, 2014, 21:53
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Michael Prinkey
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There are many who disapprove of it, but in implicit codes, you can gain the accuracy of higher-order schemes while retaining first-order stability by using deferred correction. Essentially, you build the implicit part of the transport equations using first-order upwind and then include the high-order differencing scheme explicitly as source term. You also subtract the explicit form of the first-order upwind scheme in the source term. So, as the solution converges the first-order implicit and first-order explicit contributions cancel out leaving only the high-order fluxes, yet the condition number/linear convergence properties are similar to those for first-order upwinding. There may be a hit to external (nonlinear) convergence in SIMPLE/PISO, but that is a trade off that is often worth making.

Deferred correction is used in most segregated CFD codes that rely on Gauss smoothing (say in the AMG solver)--First-order upwind keeps the coefficient matrices (generally) diagonally dominate. Implementing high-order schemes directly often requires linear solvers/preconditioners/smoothers than can handle matrices that are not diagonally dominate.
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Old   August 23, 2014, 07:19
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Simon Hahn
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Hey Michael,

at first thank you very much for your answer. At second I got another specific question about evalutating the source term and the deferred correction technique.
By using the upwind scheme, we rewrite the face values to nodal values of the grid: east => P, west => W. This gives us the (tri-diagonal) matrix.

The source term can be devided into a part for the boundary condition (SU_boundary) and the part for the difference between the high-order differencing scheme and the low-order differencing scheme (SU_s) with the values for phi from the previous time step:

SU_total = SU_boundary + SU_s with: SU_s = [phi_high-order-scheme - phi_low-order-scheme]^0

The problem now is, that my solution tends to oscillate and I don't know the reason why. As I understood, the deferred correction technique uses the values of the previous time step, applies them in the scheme for higher- and lower-order differencing scheme and refines the source term SU and consequently the calculation of phi until a predefined convergence criterium.

Thank's in advance again!

Last edited by Simon Smokes; August 23, 2014 at 12:24.
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