A well-conditioned collocation method using pseudospectral integration matrix

TL;DR

This paper introduces a new collocation method for linear differential equations that uses a specially designed polynomial basis to ensure well-conditioning and exact boundary condition enforcement, enabling stable solutions with many points.

Contribution

A novel polynomial basis based on Birkhoff interpolation is developed, leading to a well-conditioned collocation scheme with an exact inverse of the pseudospectral differentiation matrix.

Findings

Condition number is independent of collocation points.

Exact enforcement of boundary conditions.

Stable solutions with thousands of points.

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.