[FMDenaro]

Actually, the local truncation error (LTE) should be expressed somehow as:

(discrete formula) = (1/|V|) Int [S] n.vv dS + LTE


Note that you must consider the term (1/|V|) , that is 1/h^2 in 2D and 1/h^3 in 3D to have a consistent formula. If you do not consider the correct expression, the integral vanish for vanishing mesh size! That cannot define the LTE.

what you have addressed above does not define the LTE

[poliwang]

Originally Posted by truffaldino

If you consider integral over all faces of the cell (not only one face) you will get O(h^3)

I agree. Then if use the new approach to integrate overall all faces, accordingly the accuracy will also increase, i.e. from O(h^3) to O(h^4). It still says the improvement does exist!

[truffaldino]

Originally Posted by poliwang

I agree. Then if use the new approach to integrate overall all faces, accordingly the accuracy will also increase, i.e. from O(h^3) to O(h^4). It still says the improvement does exist!

Improvement exists, but its accuracy exceeds accuracy of other terms in NS.

Taking the square of average already gives the second order accuracy (I was wrong and the error after integrating over all faces is even less than O(h^3), in fact it is O(h^4) and your improvement gives O(h^5) which is the third order).

[adrin]

Though Lipo Wang is considering high resolution near the boundary to justify/explain the linear variation of u, it in no way invalidates the real question being asked. And the question is really simple. The mesh need not really be fine or adjacent to the boundary (though the boundary example clarifies the point most vividly). To this end, it is really irrelevant what the global velocity profile is (whether it's linear or parabolic).

The question relates to the assumptions made in approximating the variation of the velocity along each wall (but making sure the whole scheme is self-consistent). To this end, using just UU.n is consistent with a piecewise constant variation. For a piecewise linear variation of velocity, irrespective of the mesh density or whether it is near the boundary, Lipo Wang may indeed have a point. But, I have to urge caution. As I mentioned earlier, one has to make sure that the whole algorithm is self-consistent (in terms of discretization), so the devil's in the details.

There are many papers on high-order finite volume methods, so that would be the best place to start.

adrin

[FMDenaro]

Adrin, I agree to check if the scheme is consistent everywhere for h->0.
I would add that the improvement in the accuracy must be considered for the general FV framework.

Just to highlight some issue about FV, hoping that focus the discussion of what the improvement near or far a boundary can do...

1) U is the averaged function of u, it is a pointwise function when the volume integral is local ( h=h(x)). That means U(x)=u(x) + O(h^2) from simple Taylor expansion. Note: this is valid in the continuous framework, without using any type of discretization.

2) The order of accuracy for a FV method is evaluated from the local truncation error of the numerical flux function. You can discretize the term

(1/|V|) Int [S] n.uu dS

after a reconstruction (of some order) U=U(u) is defined and evaluate the resulting LTE. Despite the fact you can use numerical flux functions of high order, the second order approximation of U with respect to u still remain as a mathematical aspect of the integral formulation. The discretization error converges with second order slope irrespective of the numerifcal flux function. To make the averaged field U a higher order approximation, special technique must be adopted.
Some particular ENO reconstructions do that, alternatively a deconvolution technique is suitable.

I worked on this aspect for many years, so I know that the literature is very reach of papers about that. If you are interested, here is an old paper we developed:
https://www.researchgate.net/publica...-uniform_grids

you can see that using numerical flux of high order, leads to second order convergence of the discretization error. Only after inserting the deconvolutiuon, the discretization error cheanges its slope.

[poliwang]

Originally Posted by FMDenaro

Adrin, I agree to check if the scheme is consistent everywhere for h->0.
I would add that the improvement in the accuracy must be considered for the general FV framework.

Just to highlight some issue about FV, hoping that focus the discussion of what the improvement near or far a boundary can do...

1) U is the averaged function of u, it is a pointwise function when the volume integral is local ( h=h(x)). That means U(x)=u(x) + O(h^2) from simple Taylor expansion. Note: this is valid in the continuous framework, without using any type of discretization.

2) The order of accuracy for a FV method is evaluated from the local truncation error of the numerical flux function. You can discretize the term

(1/|V|) Int [S] n.uu dS

after a reconstruction (of some order) U=U(u) is defined and evaluate the resulting LTE. Despite the fact you can use numerical flux functions of high order, the second order approximation of U with respect to u still remain as a mathematical aspect of the integral formulation. The discretization error converges with second order slope irrespective of the numerifcal flux function. To make the averaged field U a higher order approximation, special technique must be adopted.
Some particular ENO reconstructions do that, alternatively a deconvolution technique is suitable.

I worked on this aspect for many years, so I know that the literature is very reach of papers about that. If you are interested, here is an old paper we developed:
https://www.researchgate.net/publica...-uniform_grids

you can see that using numerical flux of high order, leads to second order convergence of the discretization error. Only after inserting the deconvolutiuon, the discretization error cheanges its slope.

Thanks! I will go through your paper and back to you.

[adrin]

Filippo; you've expanded very nicely on what I was saying (or trying to say) when I referred to making sure that the discretization is self-consistent. High-order discretization of a parameter somewhere does not guarantee overall high-order solution (the bottleneck always being the lowest order term).

adrin

[FMDenaro]

My opinion is that approaching FV methods there is often a misunderstanding in the word "high accuracy"...the fact that U is a second order approximation to u is just a mathematical result of the integral form, is not a numerical error issue.
If we wanto to solve for the averaged velocity in the weak form, the numerical task is in computing a numerical flux function at high accuracy.