A well-conditioned collocation method using pseudospectral integration matrix

Abstract: 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 exact inverse of the pseudospectral differentiation matrix (PSDM) 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.