# How to calculate Taylor series expansion of a cell based on cell averaged derivatives

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 April 17, 2017, 04:38 How to calculate Taylor series expansion of a cell based on cell averaged derivatives #1 New Member   Manish Kumar Nayak Join Date: Jun 2015 Posts: 5 Rep Power: 11 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.

 April 17, 2017, 05:27 #2 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,831 Rep Power: 73 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?

 April 17, 2017, 05:39 #3 New Member   Manish Kumar Nayak Join Date: Jun 2015 Posts: 5 Rep Power: 11 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.

 April 17, 2017, 06:01 #4 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,831 Rep Power: 73 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..

 April 17, 2017, 06:09 #5 Senior Member   Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,831 Rep Power: 73 Have also a look to ENO/WENO reconstruction schemes, for example in the book of Leveque.

 April 17, 2017, 06:10 #6 New Member   Manish Kumar Nayak Join Date: Jun 2015 Posts: 5 Rep Power: 11 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?

 Tags cell centroid, galerkin

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