Physical motivation of Upwind scheme
 

Hello everyone,

What's the physical motivation of upwind scheme?

Best,
lnk

The fact that the exact solution of the equation d phi/dt + u d phi/dx = 0 is

phi(x,t) = phi (x- u (t-t0), t0)

that means the the solution comes from an "up-winded" 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 non-linear and diffusive case, things are more complex

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.

The rationale for upwind schemes stems from the fact that, in hyperbolic equations, information propagates at finite speeds of either positive or negative value.

For scalar equations, there is only one wave-speed and the above consideration results in using a one-sided method, in which the flux is based exclusively on the cell-value 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 "Finite-Volume Methods for Hyperbolic Problems", in which such concepts are explained very clearly.

Regards,
Francesco

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 non-vanishing 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...

That is a good point.
Actually for upwind-biased schemes "physical" and "numerical" motivations are strongly coupled. Besides the question about wave-propagation, those schemes were born as shock-capturing methods thanks to their inherent numerical dissipation, which is able to produce an entropy-satisfying, stable solution. For this reason, upwind-biased schemes are successfully used in high-speed 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. wave-propagation) to justify the adoption of upwind schemes.

For instance, the numerical properties of shock-capturing 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 subgrid-scale motions in the so-called "Implicit-LES" approach. In the latter case upwind-biased schemes are used relying exclusively upon their numerical properties, which, for some reasons, are capable to correctly reproduce the physical phenomena.

I agree ;)
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