A Highly Accurate Multi-Scale Full/Half-Order Polynomial Interpolation
Chein-Shan Liu

doi:10.3970/cmc.2011.025.239
Source CMC: Computers, Materials & Continua, Vol. 25, No. 3, pp. 239-264, 2011
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Keywords Multi-scale polynomial interpolation, Half-order polynomial interpolation, Differential quadratures, Vandermonde matrices
Abstract For the computational applications in several areas, we propose a single-scale and a multi-scale diagonal preconditioners to reduce the condition number of Vandermonde matrix. Then a new algorithm is given to solve the inversion of the resulting coefficient matrix after multiplying by a preconditioner to the Vandermonde matrix. We apply the new techniques to the interpolation of data by using very high-order polynomials, where the Runge phenomenon disappears even the equidistant nodes are used. In addition, we derive a new technique by employing an m-order polynomial with a multi-scale technique to interpolate 2m+1 data. Numerical results confirm the validity of present polynomial interpolation method, where only a constant parameter R0 needs to be specified in the multi-scale expansion. For the Differential Quadrature (DQ), the present method provides a very accurate numerical differential. Then, by a combination of this DQ and the Fictitious Time Integration Method (FTIM), we can solve nonlinear boundary value problems effectively.
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