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 nu March 7, 2002 16:47

TVD

Hi

what do we mean by a TVD schemes. is something to deal with limiter functions or the numerical solution is nonincreasing in time ?

any numerical help!

 ryoga March 7, 2002 19:02

Re: TVD

TVD stands for total variance diminishing. A good introduction into TVD schemes and flux limiters, as well, is provided in C. Laney's book, Computational Gasdynamics. Hope this helps!

Ryoga

 Paul March 7, 2002 22:33

Re: TVD

As Ryoga said, TVD means Total Variance Diminishing. In computational, it usually means that your numerical scheme won't produces more extremes than last time in the evolution, thus your result will be free of spurious oscillations. A effective means to ensure TVD is by limiter, which can be exerted in the function or its derivative. A minmod function is usually involved to kick out the too much variation.

 nu March 8, 2002 05:52

Re: TVD

i know what stand for, i've started my numerical form Toro's book. thanks paul. i.e. what i wanted to know, well done.

 Bert Laney March 11, 2002 12:50

Re: TVD

Perhaps the best way to define the "TVD" property is as follows: the sum of the maxima in the solution minus the sum of the minima does not increase in time. Strictly speaking, "TVD" does not say anything about invidual maxima or minima -- it only affects the sum total of extrema.

"TVD" is best understood as a type of stability condition. The most common symptom of numerical instability is spurious oscillation, especially of the sort that grows progressively larger with time, kind of like a car with worn shocks on a bump road. "TVD" limits the size of but does not prevent spurious oscillations. Most so-called "TVD" enforcement techniques actually ensure something much strong than "TVD," and tend to strongly discourage spurious oscillations (except possibly those that arise through interactions between characteristic variables, such as at shocks and reflective boundaries).

"TVD" is often enforced using techniques such as flux limiters, artificial viscosity, self-adjusting hybrids, etc.

A signature side-effect of "TVD" is to "clip" the peaks of maxima and minima, reducing the order of accuracy there or even inducing an artificial flat spot.

"TVD" is most often used for high-speed flows and especially shock capturing methods. A proper shock capturing method is conservative. Numerical conservation plus "TVD" implies more than either alone -- the combination strictly limits the order-of-accuracy at extrema to one.

There is obviously a great deal more to be said but hopefully this is enough for an internet chat group.

Bert Laney

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