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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 
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Manish Kumar Nayak
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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/AIAA2009605.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...derivativesi Please help me understand this. I have spent hours on the same page now. 

April 17, 2017, 05:27 

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


April 17, 2017, 05:39 

#3 
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Manish Kumar Nayak
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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) * (xx0) + avg_domain(d2P/dx2) *( (xx0)^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 
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Filippo Maria Denaro
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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 pointwise 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/dxx0 (csix0)+d2f/dx^2x0 (csix0)^2/2+... that is integrated over a general volume so that fbar(x) = 1/h Int[xh1/2,x+h2/2] f(csi) dcsi getting f_bar(x) = f(x0) + df/dxx0 [1/h Int[xh1/2,x+h2/2] (csix0) 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 cellaveraged derivatives up to degree p. Then the highorder derivatives are hierarchically limited using the cellaveraged derivatives of one degree lower. " so I suppose they introduce an approximation by substituting the pointwise derivatives with the averaged derivatives... Maybe in Ref[23] the procedure is bettere detailed.. 

April 17, 2017, 06:09 

#5 
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Filippo Maria Denaro
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Have also a look to ENO/WENO reconstruction schemes, for example in the book of Leveque.


April 17, 2017, 06:10 

#6 
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Manish Kumar Nayak
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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? 

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cell centroid, galerkin 
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