Talk:Numerical diffusion

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Mathematics rating:	Stub Class	Low Priority	Field: Applied mathematics
Please update this rating as the article progresses, or if the rating is inaccurate. Click to show/hide comments. Please add to or update the comments to suggest improvements to the article. Geometry guy 23:10, 10 July 2007 (UTC)

[edit] comments on article

I like this article, especially the green dye example.
It might be useful to explicitly state that diffusion is inherent; it is not something done by design.
"Numerical diffusion is a difficulty...": Would a specialist use the word "difficulty" in this context?
--Jtir 16:14, 8 October 2006 (UTC)

The Numerical stability article says:

• Stability is sometimes achieved by including numerical diffusion. Numerical diffusion is a mathematical term which ensures that roundoff and other errors in the calculation get spread out and do not add up to cause the calculation to "blow up".

Now I am confused. Is numerical diffusion inherent or by design? Or is the term used in more than one way? --Jtir 16:22, 8 October 2006 (UTC)

It probably needs a rework. Sometimes numerical diffusion is added deliberately to stabilize a simulation, but in general it is unavoidable in an Eulerian-style fluid simulation. I'll put something in the article... zowie 17:07, 8 October 2006 (UTC)

Thanks. That's very helpful. The lead doesn't fully capture the use of the term, because when numerical diffusion is employed deliberately it is a method to overcome other difficulties rather than a difficulty itself. Maybe the accidental and the deliberate occur simultaneously?

Here is an attempt to expand the lead sentence into a paragraph.

In computer simulations of continuous systems, such as fluids or plasmas, numerical diffusion is the propagation of errors through the approximating mesh [and with time?]. Such error propagation is inherent in discrete simulations, but it can be minimized by various methods [at the expense of tradeoffs?]. It is sometimes deliberately introduced to stabilize a simulation of a system having singularities [where are the singularities?].

Usage examples from a google search for future reference:

• "All CFD codes, whether finite-difference, finite-volume, or finite-element, suffer from the problem of numerical undershoots and overshoots in the flow variables caused by discretization of the convection term in the flow conservation equations. These problems typically occur when sharp gradients in the flow variables are encountered on the computational grid. The common solution to this problem is to add varying amounts of artificial numerical diffusion to the solution algorithm to stabilise the overall convection scheme. The diffusion has the effect of weighting the convection towards the upwind regions of the flow, hence the algorithms are referred to generically as upwinding schemes." [1]
• "Finite-difference numerical techniques based on control volumes provide the most commonly applied method for solution of heat and mass transfer problems. These first-order solutions are prone to substantial inaccuracy introduced by numerical diffusion effects. From a Lagrangian viewpoint, the inaccuracy of interpolation in space and time creates this diffusion."[2]
• "Lower resolution methods often underresolve complicated fluid interfaces spatially due to substantially increased numerical diffusion."[3]
• "The ALGR-EPCOF algorithm was designed to capture peaks and valleys and high curvature areas to within a specified error tolerance to eliminate spurious oscillation, numerical diffusion, and peak clipping."[4]

--Jtir 20:01, 8 October 2006 (UTC)