[manish_nayak]

I have been trying real hard to understand but still, I don't know how do we expand a polynomial in a Discontinuous Galerkin cell based on cell averaged derivatives.

Here is the link to the below paper: http://dept.ku.edu/~cfdku/papers/AIAA-2009-605.pdf

Also, the reference of this reference paper http://people.math.gatech.edu/~yingj...ll_new_iii.pdf Section 4.1

I have posted the same question here too:
https://scicomp.stackexchange.com/qu...-derivatives-i

Please help me understand this. I have spent hours on the same page now.

[FMDenaro]

I am not sure about your question ...do you want to express a link between the point-wise function f(x) and its averaged function fbar(x) by means of a Tayolor series?

[manish_nayak]

No, it's not really that. It's like if you have 11 nodes, i.e. a representation of 10th order polynomial in a domain, you want to write the 10th order polynomial in terms of the average of its derivatives. eg.
P(x) = avg_domain(P(x)) + avg_domain(dP/dx) * (x-x0) + avg_domain(d2P/dx2) *( (x-x0)^2 - h^2/12)) ..

I don't know how exactly this expansion is derived and those averages of derivatives are calculated. This has been taken from the papers I have posted.
Eq. 28 in 1st paper and section 4.1 in 2nd.

[FMDenaro]

Taylor and lagrangian polynomials are related ...
However, from Eq.(28) I also have doubts about the expression....As I wrote before, it expresses a link between the point-wise function and the average function and its derivatives.
Reading the paper, I suppose that first one starts in terms of the classical expansion

f(csi)=f(x0) + df/dx|x0 (csi-x0)+d2f/dx^2|x0 (csi-x0)^2/2+...

that is integrated over a general volume so that

fbar(x) = 1/h Int[x-h1/2,x+h2/2] f(csi) dcsi

getting

f_bar(x) = f(x0) + df/dx|x0 [1/h Int[x-h1/2,x+h2/2] (csi-x0) dcsi]+ ....

Now, the authors wrote that:
"First the original degree p solution polynomial within a “troubled cell” is replaced with an equivalent polynomial based on the cell-averaged derivatives up to degree p. Then the high-order derivatives are hierarchically limited using the cell-averaged derivatives of one degree lower. "

so I suppose they introduce an approximation by substituting the point-wise derivatives with the averaged derivatives...

Maybe in Ref[23] the procedure is bettere detailed..

[FMDenaro]

Have also a look to ENO/WENO reconstruction schemes, for example in the book of Leveque.

[manish_nayak]

Yes, I did read that {Ref 23} but I couldn't figure out the algorithm they used. Could you please give it a read?
Is the book Leveque finite difference methods?