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CFDtoy February 11, 2009 18:12

Higher order Gradient Calculation

Is there any specific methods to obtain higher order (>2) gradients of a variable? Gauss or Least squares approach provides 2nd order but I would require a higher order evaluation of the gradients for smooth solutions.



ganesh February 12, 2009 14:41

Re: Higher order Gradient Calculation
Dear CFDtoy,

Green--Gauss and Least squares procedures can be used to obtain higher order derivatives too. As an example, write down in 1d the Taylor expansion for the value at some point j, about the value at point i. This would include derivatives of all orders. All you need to do is to consider the error between the known (computed) value at 'j' and the truncated Taylor expansion of 'j' and minimize the sum of squares of the error over the stencil. The truncated series must be upto the order of the highest derivative you desire. Use of LS procedure for higher order derivatives includes a larger stencil of support points (to solve the overdetermined system of equations) in addition to having a larger matrix to be inverted. For turbulent flow applications, you must take special care in using such a methodology as the geometric matrix can be reasonably ill-conditioned. You can also use Green-Gauss theorem in a similarly appropriate fashion for obtaining higher derivatives (in fact in a recursive manner).

Hope this helps



CFDtoy February 12, 2009 18:00

Re: Higher order Gradient Calculation
Hello Ganesh, Thanks for the explanation. Could you kindly direct me to some algorithms to implement it (using Gauss theorem) - any papers / notes using such a higher order estimation routine would be very helpful.

Thanks !


Timon February 13, 2009 05:42

Re: Higher order Gradient Calculation
Are you looking at structured or unstructured grids? Especially on unstructured grids the (recursive) Green-Gauss procedure is inconsistent and not very accurate. The gradients will not be smooth, which will be come increasingly problematic for the higher order derivatives you find using recursive application.

Although may be not specifically applicable to your case, have a look at the articles by Svärd or Hasselbacher (see reference example below). you can at least identify some issues you might encounter and find some good follow-up references.

Magnus Svard, Jan Nordstrom, Stability of finite volume approximations for the Laplacian operator on quadrilateral and triangular grids, Applied Numerical Mathematics, Volume 51, Issue 1, October 2004, Pages 101-125, ISSN 0168-9274, DOI: 10.1016/j.apnum.2004.02.001.

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