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July 15, 2012, 12:12 
Flux limiter and explicit method CFL restriction

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Christine Darcoux
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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 evenorder upwind schemes have a two times wider stability interval than oddorder 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 

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The minmod limiter is just a simple switch between the BeamWarming and the LaxWendroff method. Both schemes are stable for clf < 2.


July 17, 2012, 14:00 

#3  
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Christine Darcoux
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Quote:
Isn't this LaxWendroff 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 (BeamWarming ?) 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 
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Filippo Maria Denaro
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I remember something concerning thirdorder upwind in this old paper:
http://www.sciencedirect.com/science...45793091900116 maybe can help you 

July 19, 2012, 12:16 

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This is slightly offtopic :
I am trying to do a Von Neumann analysis to show that the BeamWarming (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  
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Filippo Maria Denaro
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Quote:
J. C. Tannehill, Hyperbolic and hyperbolicparabolic systems, in Handbook of Numerical Heat Transfer, W. J. Minkowycz, E. M. Sparrow, G. E. Scheider 

Tags 
flux correction, stabilty, upwind diffrence 
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