# numerical error

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 March 30, 2001, 11:24 numerical error #1 bal Guest   Posts: n/a is it possible to compensate for error of coarse mesh (large cell size) by having higher order terms for discretizing the N-S equations

 March 31, 2001, 02:26 Re: numerical error #2 John C. Chien Guest   Posts: n/a (1). If you retain more higher order terms, then these higher order terms will need more mesh points to approximate them. (2). That means, in the same area, you will have to refine the mesh. This will become more difficult near a boundary.

 April 1, 2001, 09:09 Re: numerical error #3 Sebastien Perron Guest   Posts: n/a Usually, Higher Order interpolation functions will give you more precision. But in order to construct these functions, you'll need more points. As an exemple, with finite elements or FVEM, If you compute two solutions with the same number of degrees of freedom (unknowns), one with P1 functions and the other with P2 functions, the one with P1 functions sould be more dependant on the mesh (more error prone). But, These results only hold if the stability conditions for both scheme are satisfied.

 April 3, 2001, 09:39 Re: numerical error #4 Peter Young Guest   Posts: n/a For a given mesh, it's not easy to judge if the HOD schemes give better results - take centeral difference as example, the Tayler expansion indicates that it's second order accurate which really means that if you refine your mesh, the numerical error will go down much faster than the first order scheme. If your mesh is reasonably fine, then HOD schemes normally give more accurate results.

 April 5, 2001, 14:09 Re: numerical error #5 alexy Guest   Posts: n/a Solution matrix bandwidth of higher order methods can be contained and kept the same as that of lower order methods. Please check out a paper by A.Kolesnikov and A.J.Baker "Efficient implementation ... " available at http://cfdlab.engr.utk.edu/html/publ...ns/public.html in postscript. The theory has been extended to NS fomulations (look in dissertations section on the same web page).

 April 5, 2001, 14:26 Re: numerical error #6 John C. Chien Guest   Posts: n/a (1). Thank you. (2). I think you are probably talking about the finite element method, by just looking at the authors names.

 April 5, 2001, 14:50 Re: numerical error #7 alexy Guest   Posts: n/a The analysis is performed on an assembled fe stencil, which makes it applicable to fd formulations.

 April 5, 2001, 15:22 Re: numerical error #8 kalyan Guest   Posts: n/a A question related to this issue. If you want higher order discretization but low matrix band width, do you need to solve more equations (more dependent variables) than in the method with higher order discretization and higher band width. If this is the case, you have to choose between solving a matrix with high band width or a bigger matrix with low band-width. Is this a correct assesment ? In finite difference methods, compact schemes allow for higher order accuracy by solving low band-width matrices. However, the integration has to be implicit to retain efficiency.

 April 5, 2001, 15:30 Re: numerical error #9 alexy Guest   Posts: n/a You have to solve more equations if you use traditional fe methodology when additional test/trial functions are introduced in order to increase the order of accuracy (quadratic, cubic bases for example). The paper mentioned above uses a very different ideas and not only the matrix bandwidth, but also the number of equations to solve is the same as in lower order methods. One therefore obtains high order (lower truncation error) solution at the same computational expense as a lower order solution.

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