Fast construction of the discrete Green operator for a second order ordinary differential equation

TL;DR

This paper presents a fast algorithm for constructing the discrete Green matrix for second order ODEs using spectral collocation and discrete cosine transforms, enabling efficient solutions for high-degree discretizations.

Contribution

The authors develop a novel, efficient method to construct the discrete Green matrix for second order ODEs using spectral collocation and DCT, facilitating large-scale computations.

Findings

Green matrix construction is accelerated using spectral collocation and DCT.

The method allows matrix-free application of the Green matrix.

Efficient for high polynomial degrees and large numbers of collocation points.

Abstract

We consider linear second order differential equation y''= f with zero Dirichlet boundary conditions. At the continuous level this problem is solvable using the Green function, and this technique has a counterpart on the discrete level. The discrete solution is represented via an application of a matrix -- the Green matrix -- to the discretised right-hand side, and we propose an algorithm for fast construction of the Green matrix. In particular, we discretise the original problem using the spectral collocation method based on the Chebyshev--Gauss--Lobatto points, and using the discrete cosine transformation we show that the corresponding Green matrix is fast to construct even for large number of collocation points/high polynomial degree. Furthermore, we show that the action of the discrete solution operator (Green matrix) to the corresponding right-hand side can be implemented in a matrix-free fashion.