Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jan Blechta, Vít Průša, Ladislav Trnka, Karel Tůma

TL;DR

The paper addresses solving the boundary-value problem $y''=f$ with zero Dirichlet conditions by constructing a discrete Green operator for spectral collocation. It develops a fast, exact method to form the Green matrix $m G_N$ using Chebyshev--Fourier series and discrete cosine transforms, enabling a matrix-free application of the discrete solution operator to the right-hand side. Key contributions include closed-form integral expressions for Green-matrix entries, a clear interpretation of $m G_N$ as a left and/or right inverse to the second-derivative operator under appropriate boundary- and degree-extensions, and practical Matlab/Mathematica implementations with Chebfun interoperability. The approach scales to large collocation sets, offers a direct discrete solution operator, and provides a computationally efficient alternative to solving the linear system, with careful attention to discrete scalar products and centrosymmetry of the differentiation matrix.

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.

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

TL;DR

The paper addresses solving the boundary-value problem with zero Dirichlet conditions by constructing a discrete Green operator for spectral collocation. It develops a fast, exact method to form the Green matrix using Chebyshev--Fourier series and discrete cosine transforms, enabling a matrix-free application of the discrete solution operator to the right-hand side. Key contributions include closed-form integral expressions for Green-matrix entries, a clear interpretation of as a left and/or right inverse to the second-derivative operator under appropriate boundary- and degree-extensions, and practical Matlab/Mathematica implementations with Chebfun interoperability. The approach scales to large collocation sets, offers a direct discrete solution operator, and provides a computationally efficient alternative to solving the linear system, with careful attention to discrete scalar products and centrosymmetry of the differentiation matrix.

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.

Paper Structure

This paper contains 20 sections, 103 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Fundamentals of spectral collocation method
  3. Standard approach
  4. Green function based approach
  5. Green matrix construction
  6. Chebyshev--Fourier series and primitive functions
  7. Chebyshev--Fourier series, node values, and the discrete cosine transform
  8. Discrete cosine transform---Wolfram Mathematica
  9. Discrete cosine transform---Matlab
  10. Computation of $\int_{\xi=-1 }^{x_k} l_i(\xi) \, \mathrm{d} \xi$ and $\int_{\xi=x_k }^{1} l_i(\xi) \, \mathrm{d} \xi$
  11. Evaluating Chebyshev polynomials at collocation points
  12. Algorithm
  13. Green matrix as an inverse matrix to the second order spectral differentiation matrix
  14. Left inverse
  15. Right inverse
  16. ...and 5 more sections

Figures (2)

  • Figure 1: Green function $g(x, \xi)$ for problem $\frac{\mathrm{d}^2 {y}}{\mathrm{d} {x}^2} = f$; interval $\left(-1, 1\right)$, zero Dirichlet boundary conditions.
  • Figure 2: Integrands in \ref{['eq:26']}, $i=2$, $N=11$.