Flux limiter and explicit method CFL restriction
 

A class of TVD scheme was developed by Sweby[1] where a flux limiter is added to the Second Order Upwind (SOU) schemes differencing scheme to prevent the formation of oscillations in the scalar field.

I am interested by the CFL restriction of these scheme in the context of the explicit forward euler time integration.

One important property of the SOU discussed by Leonard [2] is that even-order upwind schemes have a two times wider stability interval than odd-order ones. Thus, SOU is stable at the extended interval 0 < CFL < 2.

Question : Are there any TVD scheme based on SOU that also preserve stability for CFL < 2 or more ?

Thanks for your help !

Christine

[1] P. K. Sweby. High resolution schemes using flux limiters for hyperbolic conservation laws.
SIAM Journal of Numerical Analysis, 21(5):995–1011, 1984.

[2] Leonard, B. P. Stability of explicit advection schemes. The
balance point location rule.
Int. J. Numer. Meth. Fluids 38, 471 –514, 2002.

The minmod limiter is just a simple switch between the Beam-Warming and the Lax-Wendroff method. Both schemes are stable for clf < 2.

Thanks for your answer, but I am a little confused ...

Isn't this Lax-Wendroff equivalent to the central difference scheme (phi=1 in the Sewby diagram) ? It is only stable for clf<=1 as far as I know.

As the SOU (Beam-Warming ?) is given by phi=r in the diagram and is TVD up to phi=2, I think that a limiter of the form

min(r, something smaller or equal to 2)

should be a good candidate.

This is slightly off-topic :

I am trying to do a Von Neumann analysis to show that the Beam-Warming (yes, this is another name for the SOU) is stable for μ < 2. Maybe I am wrong, but I only get the classical CFL criterion μ < 1.

Could someone point me a reference where I could find the details of the analysis for μ < 2 ?

I remember Von Neumann stability analysis of several schemes on
J. C. Tannehill, Hyperbolic and hyperbolic-parabolic systems, in Handbook of Numerical Heat Transfer, W. J. Minkowycz, E. M. Sparrow, G. E. Scheider