A well-conditioned collocation method using a pseudospectral integration matrix
Samson, Michael Daniel
Date of Issue2014
School of Physical and Mathematical Sciences
In this paper, a well-conditioned collocation method is constructed for solving general $p$th order linear differential equations with various types of boundary conditions. Based on a suitable Birkhoff interpolation, we obtain a new set of polynomial basis functions that results in a collocation scheme with two important features: the condition number of the linear system is independent of the number of collocation points, and the underlying boundary conditions are imposed exactly. Moreover, the new basis leads to an exact inverse of the pseudospectral differentiation matrix of the highest derivative (at interior collocation points), which is therefore called the pseudospectral integration matrix (PSIM). We show that PSIM produces the optimal integration preconditioner and stable collocation solutions with even thousands of points.
SIAM journal on scientific computing
© 2014 Society for Industrial and Applied Mathematics. This paper was published in SIAM Journal on Scientific Computing and is made available as an electronic reprint (preprint) with permission of Society for Industrial and Applied Mathematics. The paper can be found at the following official DOI: [http://dx.doi.org/10.1137/130922409]. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper is prohibited and is subject to penalties under law.