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August 6, 2017, 04:24 
Understanding Davis artificial viscosity

#1 
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Oleg Sutyrin
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I'm solving 2D Euler's equations in Cartesian coordinates:
using finite difference MacCormack method with Davis artificial viscosity which dampens nonphysical oscillations. The MacCormack method is pretty simple and not shown here. Davis viscosity works good in my simulations, but is quite difficult; I'm trying to fully understand it in order to apply it cylindrical coordinates later. I hope that some of you may help me with unclear parts of it (shown in bold font below). Viscous terms are added after 2nd MacCormack step as follows (formulated in 1D for simplicity): where denote field values (at th coordinate point and th time layer) calculated by MacCormack scheme as usual, and , are forward and backward differences of U with nonlinear coefficients: Where is a limited coefficient based on Courant number: is the Jacobian matrix of vector , is it's spectral radius (for Euler's equations it equals , where is the speed of sound). is simple limiting function: And finally , are slope ratios: Where is scalar product. These ratios are positive in monotone areas and negative in nonmonotone areas, so that in general coefficients like are equal 2 in nonmonotone areas; about 1 in monotone, but "curvy" areas; and equal 0 in monotone (and rectilinear) areas. If both these () coefficients are equal 2, and both are equal 0.25 we have an effective second derivative of the field value: But where is the division by ? If the viscosity would be simply the 2nd derivative, the scheme would look like Is effectively equals ? And, what if is nonzero and is zero? Does it still yield an effective 2nd derivative? 

August 6, 2017, 05:10 

#2 
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Filippo Maria Denaro
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I think you should see the artificial viscosity as a term that disappears as dx>0. Therefore, multiply and divide by dx^2 and you get a second order derivative multiplied by dx^2.
You wrote dt/dx^2 but the dimensions are no longer congruent 

August 6, 2017, 06:00 

#3  
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Oleg Sutyrin
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Thanks for such a fast answer, FMDenaro!
Quote:
But spurious oscillations do not disappear as . If artificial viscosity disappears as , how would it dampen the oscillations then? I can't say that I understand that completely. Could you elaborate? 

August 6, 2017, 06:10 

#4 
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Filippo Maria Denaro
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The artificial viscosity has nothing to do with the physical viscosity. It is just an added terms that has the aim of dumping oscillations. But it has to be consistent with the original PDE equation as h>0.
I suggest a reading in the fundemental textbooks about the numerical solution of Euler equations. 

August 6, 2017, 06:20 

#5 
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Oleg Sutyrin
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Yeah, it seems that deep diving into the theory is inevitable :)
Thanks again! 

August 7, 2017, 14:54 

#6 
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Martin Hegedus
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As a general statement, artificial viscosity is a consequence of numerical differentiation. It is not a physical term. It can be determined, in a hand waving way, from the wave equations. In a nutshell, central difference is a result of taking the limit when coming from the right and left.
In a sense, dF/dx is a function of lim (+ and )>0 [dF/dx(+) + dF/dx()] + lim (+ and )>0 [dF/dx(+)  dF/dx()]. If everything was perfect, and assuming no discontinuities, [dF/dx(+)  dF/dx()] would be zero. But numerically it is not. Thus the C*(U(i+1)2*U(i)+U(i1)) where C is a knob. 

August 12, 2017, 05:34 

#7  
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Oleg Sutyrin
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Quote:
"Knob" has too many meanings in English language, so I can't understand the meaning of this phrase :) 

August 12, 2017, 07:21 

#8 
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Filippo Maria Denaro
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I suggest having a reading to the texbook of Leveque (for example pag.7172) to start with the concept of artificial viscosity explicitly added to the discretization in order to dump oscillations and numerical viscosity present in the local truncation error that is implicitly induced by the discretization.


August 12, 2017, 07:33 

#9  
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Oleg Sutyrin
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Quote:


August 12, 2017, 07:37 

#10 
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Filippo Maria Denaro
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August 12, 2017, 07:57 

#11  
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Oleg Sutyrin
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