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High precision computations of multiquadric collocation method for partial differential equations

Multiquadric collocation method is highly efficient for solving partial differential equations due to its exponential error convergence rate. More amazingly, there are two ways to reduce the error: the traditional way of refining the grid, and the unexpected way of simply increasing the value of shape constant $c$ contained in the multiquadric basis function, $sqrt{r^2 + c^2}$. The latter is accomplished without increasing computational cost. It has been speculated that in a numerical solution without roundoff error, infinite accuracy can be achieved by letting $c
ightarrow infty$. The ability to obtain infinitely accurate solution is limited only by the roundoff error induced instability of matrix solution with large condition number. Using the arbitrary precision computation capability of {it Mathematica}, this paper tests the above conjecture. A sharper error estimate than previously obtained is presented in this paper. A formula for a finite, optimal $c$ value that minimizes the solution error for a given grid size is obtained. Using residual errors, constants in error estimate and optimal $c$ formula can be obtained. These results are supported by numerical examples.

Identiferoai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0614106-030140
Date14 June 2006
CreatorsLee, Cheng-Feng
ContributorsChien-Sen Huang, Zi-Cai Li, Leevan Ling, ALLEN T. L. HORNG, Tzon-Tzer Lu
PublisherNSYSU
Source SetsNSYSU Electronic Thesis and Dissertation Archive
LanguageEnglish
Detected LanguageEnglish
Typetext
Formatapplication/pdf
Sourcehttp://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0614106-030140
Rightscampus_withheld, Copyright information available at source archive

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