# Flux limiter and explicit method CFL restriction

 Register Blogs Members List Search Today's Posts Mark Forums Read

 July 15, 2012, 12:12 Flux limiter and explicit method CFL restriction #1 New Member   Christine Darcoux Join Date: Jul 2011 Posts: 15 Rep Power: 14 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.

 July 17, 2012, 12:06 #2 New Member   Join Date: Jul 2012 Posts: 7 Rep Power: 13 The minmod limiter is just a simple switch between the Beam-Warming and the Lax-Wendroff method. Both schemes are stable for clf < 2.

July 17, 2012, 14:00
#3
New Member

Christine Darcoux
Join Date: Jul 2011
Posts: 15
Rep Power: 14
Quote:
 Originally Posted by bigorneault The minmod limiter is just a simple switch between the Beam-Warming and the Lax-Wendroff method. Both schemes are stable for clf < 2.

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.

 July 17, 2012, 15:03 #4 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,764 Rep Power: 71 I remember something concerning third-order upwind in this old paper: http://www.sciencedirect.com/science...45793091900116 maybe can help you

 July 19, 2012, 12:16 #5 New Member   Join Date: Jul 2012 Posts: 7 Rep Power: 13 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 ?

July 19, 2012, 13:30
#6
Senior Member

Filippo Maria Denaro
Join Date: Jul 2010
Posts: 6,764
Rep Power: 71
Quote:
 Originally Posted by bigorneault 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

 Tags flux correction, stabilty, upwind diffrence