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August 18, 2012, 09:53 
Physical motivation of Upwind scheme

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lnk
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Hello everyone,
What's the physical motivation of upwind scheme? Best, lnk 

August 18, 2012, 10:46 

#2  
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Filippo Maria Denaro
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phi(x,t) = phi (x u (tt0), t0) that means the the solution comes from an "upwinded" region of the space. Than cna be seen also in terms of the characteristics, the solution is constant along the trajectories. For 3D linear case the solution is equivalent, but when you consider the nonlinear and diffusive case, things are more complex 

August 23, 2012, 11:27 

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The idea behind upwind schemes states that the solution would essentially depends on what happened in upwind locations as Filipo stated it. But to my mind the use of upwind scheme is more dictated by stability numerical considerations rather than physical ones. For hyperbolic equations where the solution depends only from upwind locations characteristic method (based on this upwind dependence) is an option but not the only one. On elliptic equations where the solution on one point depends on every locations, one use upwind scheme in order to stabilize the flow, due to the amount of numerical diffusion (artificial viscoisty) that such scheme brings in the system. 

September 3, 2012, 05:23 

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Francesco Capuano
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For scalar equations, there is only one wavespeed and the above consideration results in using a onesided method, in which the flux is based exclusively on the cellvalue from which the information is coming. For a system of n equations, we have n waves travelling at n speeds, and there may be waves coming from both directions. In this case, a characteristic decomposition is needed in order to obtain a decoupled system of scalar equations to which apply the upwind method, before going back to the physical space. I suggest the book of R. J. Leveque "FiniteVolume Methods for Hyperbolic Problems", in which such concepts are explained very clearly. Regards, Francesco 

September 3, 2012, 06:45 

#5 
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Filippo Maria Denaro
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I think that talking about "physical motivation" has to be better focused ...
Using ideal fluid the system of Euler equations is hyperbolic but "Real physics" in fluid dynamics involves real fluid, with nonvanishing molecular viscosity (and other transport coefficients) that makes the governing system of NS equations hyperbolic in the continuity but parabolic in the momentum and energy equations. That makes much more problematic the concept of physical motivation of upwind, I think that in such a case is better to say "numerical motivation" in using upwind schemes. Theory for parabolic equations do not justify specific direction in waves propagation as for pure hyperbolic equation. For very small viscosity one can consider a perturbation approach on the hyperbolic system. The issues become complex... Just as an example, DNS and especially LES, are quite always performed avoinding upwind discretizations... 

September 3, 2012, 08:44 

#6  
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Francesco Capuano
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Actually for upwindbiased schemes "physical" and "numerical" motivations are strongly coupled. Besides the question about wavepropagation, those schemes were born as shockcapturing methods thanks to their inherent numerical dissipation, which is able to produce an entropysatisfying, stable solution. For this reason, upwindbiased schemes are successfully used in highspeed aerodynamics, where strong shocks are present. In this case, such schemes work very well thanks both to a numerical and a physical reason. In many other cases (e.g. parabolic equations), as Filippo states, there might be no particular physical arguments (i.e. wavepropagation) to justify the adoption of upwind schemes. For instance, the numerical properties of shockcapturing methods make them unsuitable for calculations in which small numerical viscosity is needed (e.g. "classical" LES), but their dissipation can be properly used to mimic subgridscale motions in the socalled "ImplicitLES" approach. In the latter case upwindbiased schemes are used relying exclusively upon their numerical properties, which, for some reasons, are capable to correctly reproduce the physical phenomena. 

September 3, 2012, 09:55 

#7  
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Filippo Maria Denaro
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Quote:
for this reason the "physical/numerical motivations" for using upwind schemes should be seen in a framework in which the physical problem, the class of solution, the goal of the simulation are taken into account 

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