In a paper entitled "The ULTIMATE conservative difference scheme", Leonard states that "Sweby's TVD region is grossly over-restrictive, resulting in predicted (normalised) face values which are too small". Can someone discuss this in the context of stabilty and accurate solutions? Is Leonard suggesting that his ULTIMATE strategy supercedes Sweby's TVD ideas? Is the ULTIMATE strategy TVD? (I don't think so).

Thank you.

Hi,

I made some comparisons with 2D triangular finite volumes and there was a lot of numerical diffusion. Could you post the papers, you're reffering?

There are no ULTIMATE conservative difference scheme. Too strong words

"The ULTIMATE conservative difference scheme applied to unsteady one-dimensional advection". In Computer Methods in Applied Mechanics and Engineering 88 (1991) 17-74 by B.P. Leonard.

Heya,

Leonard has always been a bit obsessed with differencing schemes. The big contribution here is the NVD (Normalized Variable Diagram) a few papers before hand. Here, he offers a novel criterion for bounding variables, which is less restrictive and can be used in place of TVD. As a reminder, the TVD criterion is derived from the entropy condition.

NVD looks at the solution in the normalised space and indicates what action should be taken and when. There is, obviously, quite a lot of lattitude in what you are allowed to do (make your own NVD scheme) and the whole of TVD is a subset of the NVD space. I heartily recommend the papers, they are reasonbly easy to read.

The ULTIMATE (I think this is an acronym, plese check the paper) scheme by leonard gives additional sensitisation of the scheme to time-step/Co number + the paper includes a number of various-order bounded schemes. I don't think it is intended as "the end of all convection discretisation research", but I consider the papers interesting.

Enjoy,

Hrv

P.S. For my very own NVD scheme please see...

Thanks Hrvoje. If the TVD scheme is a subset of the NVD (as can be seen at the end of the paper I referenced earlier), then that means the NVD scheme is not always TVD. Indeed, it is mostly not TVD in its practical use. Is this correct? Also, how then do we ensure an oscillation free problem? I know TVD does not necessarily ensure an oscillation free problem, but in conjunction with positivity and mass conservation, we end up with a generally oscillation free flow. What is the criteria for oscillation free flow when using TVD?

Thanks for your remarks.

NVD scheme is not always TVD: correct. That was the point, as TVD was considered too diffusive.

How then do we ensure an oscillation free problem: we look at the shape of the solution and for the special case of convection-oriented profile with high second gradient we blend down from the second-order scheme to a bounded first order scheme. If an oscillation occurs, NVD clearly says to switch to upwinding (unconditionally bounded).

TVD uses the entropy condition to preserve boundedness; NVD just looks at the solution. This means that NVD is sometimes a bit "late" in fixing the problem (it has got to identify it first), but in practice this is good enough.

Hrv

Thanks again.

"If an oscillation occurs, NVD clearly says to switch to upwinding": Is this is in effect what you mean by switching to 1st order?

How does TVD use the entropy condition to ensure boundedness?

I have seen alot about TVD in theory, but never seen how to ensure some scheme is TVD. The only thing I have seen is the the positivity condition which apparently is a stronger condition than TVD, but is nearly always the method used to wnusre TVD. Can you elaborate on this?

Thanks again for you help.

TVD schemes do not necessarily guarantee entropy consistency. On the other hand monotone schemes are entropy consistent but are unfortunately only first order accurate.

How I can I check something for Entropy consistency? I don't really find explainations in books of much use...