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Old   December 18, 2009, 19:14
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Dear Friends,

I was looking at Jasak's paper on NVD differencing scheme for arbirary unstructured meshes. The method is quite attractive and blends the UD and CD scehmes to acheive a higher order bounded scheme. In its implementation, there is a term that reads

\tilde \phi_c = 1 - \frac{phi_c - phi_d}{\nabla \phi_c \cdot d}

where the second term is essentially a ratio of cell to face gradients. In regions of smooth flow, the gradeints can be numerically quite small, so it is possible that this ratio (being eps1/eps2 form) can be greater than 1 and hence \tilde \phi_c becomes less than 0. The NVD scheme recommends that the UD be used in case \tilde \phi_c becomes lesser than 0, in which case we have a lower accuracy in a region, where we otherwise would expect a CD scheme to work without problems. Is there a better numerical workaround for this case other than say setting the scheme to be CD if the gradient has value less than some eps tolerance ?

P.S.: I have seen these problems with limiters too, where they becopme active in smooth flow regions (such as freestream for external flows) and this case here is no different since conventional limiters can also be brought into NVD form.

Thanks and Regards,

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Old   December 18, 2009, 20:26
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The main purpose of high resolution schemes (HRS) is to obtain bounded results. If it shifts to CD, it may cause under/overshoots which violates boundedness criteria. so almost all HRS (except SHARP of Leonard or SECBC of Tao), in such regions FOU is used.
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Old   December 18, 2009, 21:38
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Dear Meda,

That is right, though I am interested in smooth flow regions where CD satifies boundedness, yet the Gamma-NVD scheme might revert to UD, purely because of ratio of two small quantities becoming O(1). I have seen this happen with slope limiters in freestream, where they become active when there is no reason to expect this behaviour. I am looking if there is a nice "fix" to resolve this issue.


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Old   April 11, 2012, 11:38
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I don't understand why you wanna use NVD scheme while actually you still want CD in smooth field, why not just use CD throughout the domain if your case allow you to do so.
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