my lecturer says the scheme he uses if diffusion free.

what does it mean by diffusion free, and which schemes are and which are not diffsion free?

also he seemed to imply it had something to do with the type of mesh?

can some one please explain. thanks

i guess by diffusion free he means Numerical diffusion free

The higher the grid resolution, the lower the diffusion. The higher the order of accuracy of discretization, the lower the diffusion.

As far as I know, the central difference scheme is less diffusive and upwind schemes are quite diffusive.

But is there any scheme which has no numerical diffusion ? I have never heard of.

Li

In determining the numerical behavior of a scheme you must look at order of the truncated terms (from the discretization)

There are two types of numerical error - dissipation (diffusion) and dispersion (oscillatory)

Even order truncatation tends to be dissipative Odd order truncation tends to be dispersive

First orders schemes (first order upwind, Lax scheme, etc) have second order truncation error thus they are dissipative. This is why obtaining solutions tend to be easy for first order schemes but are notoriously inaccurate unless you use VERY fine grids.

Second order schemes (central difference, second order upwind) have third order truncation error so they tend to be disperive or oscillatory. Due to this solutions can be difficult to obtain. Usually modifications are made to damp out the oscillations: artificial viscosity for central differencing and TVD and ENO schemes for second order upwinding.

The same holds true for higher order schemes. The difficulty with higher order schemes is maintaining a uniform accuracy on the boundaries (because the stencil gets large).

Hope that clears thing up - Oh and to answer your inital question NO CFD algorithm is without numerical dissipation - even the best schemes contain a little

I have two genernal questions: (1) Is really a diffusion free scheme possible? (2) What wiil be happen if we choose a high-order scheme with lower-order boundary ? I mean, for example, we use a 4-order scheme, but in boundary condition, it is only 2-order precision, what is the effect for the lower-order scheme in boundary?

Thanks your responding in advance!

As others have written before some numerical diffusion is necessary if you want to get reasonable numerical solution especially for convection-dominated systems.

It is enough if the boundary scheme is one order less accurate than the interior scheme. The overall accuracy will still be equal to that of the interior scheme. I cant recall where I have read this, probably in a paper by D. W. Zingg. Hope somebody else can point to some references.

Be careful Chris; In answer to your answer: " ... NO CFD algorithm is without numerical dissipation - even the best schemes contain a little" The Hyper-C scheme of Leonard see 1991 Comp. Meth. in Appl. Mech. Eng. Vol. 88, p.17-74 is completely diffusion free, but pretty useless as it turns any moderate gradient into a step profile.

I'm not 100% sure.. but, Hirsh's book discusses the issue on the interior/boundary accuarcy. It should have a pointer to some papers.

I kind of remember seeing a diff. free schemes in Lohner's book. It supposedly makes the dissipation term propotional to residual.

I feel like an idiot.. Time to read/study more.

In

A residual-based compact scheme for the compressible Navier-Stokes equations by Alain Lerat and Christophe Corre

a centered scheme is modified by adding dissipation that depends on the residual.

<blockquote style="border: solid 1px #006699;"> ... the process for getting a high accuracy as well as the construction of the numerical dissipation rely strongly on the residual vanishing at steady state. Starting from a basic centered scheme, the idea for increasing the accuracy order is not to cancel the leading term of the truncation error, but to add a new error term so that the resulting truncation error be expressed in terms of derivatives of the residual only. </blockquote>

Even though the residual vanishes at steady state, the scheme still has dissipation.

<blockquote style="border: solid 1px #006699;"> For simplicity and robustness, we use a first-order dissipation operator, which is compatible with any accuracy order. More precisely, by applying this operator, a centered scheme accurate at order 2p becomes a dissipative scheme accurate at order 2p-1 (dissipation at steady state comes from the truncation error of the dissipation operator). </blockquote>

I haven't come across any scheme which is totally free of dissipation. Would be interested to hear about such schemes. If you want the above paper then mail me at cpraveen[at]gmail.com

Thanks for all of your kindly reply.

Zeng