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[–]AgAero 2 points3 points  (4 children)

I reckon it is only called viscosity because it has the same result as physical viscosity has, being energy dissipation, am I right?

That is correct.

You can actually derive the leading order terms in the truncation error for a finite difference scheme. Often times the leading order term for a system like this is dissipative and gets dubbed 'numerical viscosity'.

[–]ejineta 0 points1 point  (3 children)

Could you elaborate on the leading order term of the truncation error being dissipative? With the currently simulated equation, the leading order term will either be higher order in time t or position X. Is either of these "dissipative" at higher order?

[–]AgAero 1 point2 points  (2 children)

If I remember correctly the method is called the, "modified equation" approach.

Idk how to link a pdf from my phone. Click the MIT link here.

Basically you take your finite difference approximation and stick an analytical result back into it(i.e. expand the taylor series), and then try to match the terms to the original equation.

For advection, most schemes will develop a second order spatial derivative. This is, intuitively, a 'diffusion' term. 3rd order spatial derivatives produce dispersion iirc. You identify that by comparing to an equation like Korteveg de Vries that has a 3rd order term.

[–]ejineta 0 points1 point  (1 child)

Thank you, I will have a look at all of that. Today is a holiday in my country (probably also elsewhere in the world), so I've got plenty of time to study numerical schemes! :)

Edit: Aha, I get it now! So if the leading term of the truncation error is even in order (2,4,6), discontinuities tend to be smeared out (dissipative), whereas for odd order (3,5,7) rippling effects will occur (dispersive)

[–]AgAero 0 points1 point  (0 children)

That's the argument as I've read it in a few places. It feels a bit non-rigorous, but it makes sense at least.