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

作者：Liu Chein Shan^*

来源：Computers Materials & Continua, 2011, 25(3): 239-263.

• Multi-scale polynomial interpolation
• Half-order polynomial interpolation
• Differential quadratures
• Vandermonde matrices

For the computational applications in several areas, we propose a singles-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 R(0) 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.