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Central schemes like JST uses artificial dissipation for the stabilization. This modification is an artificial one. Does this additional term makes system inconsistent? Can we expect this term to be zero at the end of the simulation? This is in reference with incompressible flow schemes. In the following references they have used the pressure differences as a dissipation to continuity equations. Is it must to reach these terms to zero at the end of the simulation (steady)?

In principle for Artificial compressibility method lower values of norms of time derivative terms indicates the satisfaction of divergence of velocity. But in the following cases where artificial dissipation is added to continuity equation which will not go zero at the end of pseudo-transience. Will there also norms of time derivative terms indicates the satisfaction of divergence of velocity?

Reference 1(Link: http://heja.szif.hu/ANM/ANM-030110-A/anm030110a.pdf) http://heja.szif.hu/ANM/ANM-030110-A/anm030110a.pdf

Reference 2(Link: http://www.sciencedirect.com/science/article/pii/S0021999199963155) http://www.sciencedirect.com/science/article/pii/S0021999199963155

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    $\begingroup$ Not sure I understand the question correctly, but typically an artificial diffusion term will go to zero as the resolution is made finer and dx and dt go to zero. This makes the scheme consistent. The term is not zero at the end of the simulation, however. $\endgroup$ – Daniel Oct 30 '15 at 8:04
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    $\begingroup$ I think it would be useful if you wrote down the formulas for the scheme, since not everyone is immediately familiar with these methods. $\endgroup$ – Wolfgang Bangerth Oct 30 '15 at 19:52
  • $\begingroup$ @DanielRuprecht I have added references in my question. Thanks! $\endgroup$ – Shainath Oct 31 '15 at 6:33
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Perhaps your query is regarding "artificial compressibility" vs "artificial dissipation/viscosity". Artificial compressibility method is a way to solve the incompressible Navier-Stokes equations where the divergence free condition is satisfied by doing "inner iterations". This is an alternative to projection methods. It is unrelated to the artificial viscosity that appears (or is added) in solving the hyperbolic equation. Artificial dissipation helps in stabilizing the advection scheme and is not related to the pseudo transient continuance of the artificial compressibility method. The extra dissipation term will not be zero at the end of the inner iterations which is normal. Pletcher, Tannenhill and Anderson have a section which discusses this issue.

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  • $\begingroup$ In first reference JST scheme is applied on the framework of artificial compressibility. My query is related to artificial dissipation only. $\endgroup$ – Shainath Nov 25 '15 at 6:45
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    $\begingroup$ Fair enough. I see what you mean now. I suspect what you are asking has something to do with co-located variables and the checker-boarding problem associated with that. Pletcher, Tannenhill and Anderson have a section which discusses this issue. $\endgroup$ – Abhilash Reddy M Nov 25 '15 at 17:35
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All of the schemes I have seen are consistent. To be concrete, the first reference essentially rewrites their original equation

$$ \frac{\partial U}{{\partial t}} + \nabla H(U) = 0$$

into

$$ \frac{\partial U}{{\partial t}} + \nabla H(U) + h^3 \left( \frac{\partial}{\partial x} (\epsilon \frac{\partial ^3}{\partial x^3} U) + \frac{\partial}{\partial y} (\epsilon \frac{\partial ^3}{\partial y^3} U) \right)= 0$$

Unless $U$ is badly behaved (e.g. there is a shock which creates a discontinuity in $U$), as $h$, the mesh spacing, goes to zero, we recover the original equation. In that way, it behaves like other resolution dependent artifacts (e.g. approximating derivatives with finite differences). To ensure that it does not cause problems, you have to run the model at different resolutions.

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