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Filtering the Tau method with Frobenius-Padé Approximants

João Carrilho de Matos, José M. A. Matos, Maria João Rodrigues

TL;DR

This work targets the reduced convergence of spectral methods near singularities by post-processing Tau-method solutions with Frobenius-Padé approximants derived from orthogonal series. By constructing Chebyshev- or Legendre-Padé filters from Tau coefficients and using a Froissart-table diagnostic to avoid doublets, the authors show improved accuracy and the ability to estimate singularities via Padé poles. The paper provides both explicit formulas for Chebyshev- and Legendre-Padé approximants, and demonstrates through numerical experiments that careful filter selection yields stable, higher-fidelity approximations even when the exact solution has nearby branch points. The approach thus extends the effectiveness of spectral/Tau methods in the presence of near-domain singularities and offers practical tools for singularity localization.

Abstract

In this work, we use rational approximation to improve the accuracy of spectral solutions of differential equations. When working in the vicinity of solutions with singularities, spectral methods may fail their propagated spectral rate of convergence and even they may fail their convergence at all. We describe a Padé approximation based method to improve the approximation in the Tau method solution of ordinary differential equations. This process is suitable to build rational approximations to solutions of differential problems when their exact solutions have singularities close to their domain.

Filtering the Tau method with Frobenius-Padé Approximants

TL;DR

This work targets the reduced convergence of spectral methods near singularities by post-processing Tau-method solutions with Frobenius-Padé approximants derived from orthogonal series. By constructing Chebyshev- or Legendre-Padé filters from Tau coefficients and using a Froissart-table diagnostic to avoid doublets, the authors show improved accuracy and the ability to estimate singularities via Padé poles. The paper provides both explicit formulas for Chebyshev- and Legendre-Padé approximants, and demonstrates through numerical experiments that careful filter selection yields stable, higher-fidelity approximations even when the exact solution has nearby branch points. The approach thus extends the effectiveness of spectral/Tau methods in the presence of near-domain singularities and offers practical tools for singularity localization.

Abstract

In this work, we use rational approximation to improve the accuracy of spectral solutions of differential equations. When working in the vicinity of solutions with singularities, spectral methods may fail their propagated spectral rate of convergence and even they may fail their convergence at all. We describe a Padé approximation based method to improve the approximation in the Tau method solution of ordinary differential equations. This process is suitable to build rational approximations to solutions of differential problems when their exact solutions have singularities close to their domain.

Paper Structure

This paper contains 16 sections, 8 theorems, 55 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Operational Tau Method
  3. Improved operational Tau method
  4. Numerical example
  5. Error analysis in the Tau Method
  6. Errors on the coefficients in example \ref{['exampleSQRT']}
  7. Frobenius-Padé approximants from orthogonal series
  8. Chebyshev Polynomials:
  9. Legendre Polynomials:
  10. The Filtering Process
  11. Error in the filtering process
  12. The Froissart table
  13. Estimation of singularities
  14. Algebraic properties of filters
  15. Numerical example
  16. ...and 1 more sections

Key Result

Proposition 1

Let $p,q\in \mathbb{N}_0$, and If $\mathbf{H}^{[p/q]}$ is nonsingular then are determined by

Figures (7)

  • Figure 1: Rate of consecutive errors $\| e_{n+1}\|_{w}/ \|e_{n}\|_{w}$, for $n=9,10,\ldots,149$.
  • Figure 2: Error curves $y(t)-y_n(t)$ and error curves approximations $\tilde{e}_{n+1}(t)$ and $\tilde{e}_{n+20}(t)$ obtained with the Tau method for example \ref{['exampleSQRT']}.
  • Figure 3: Froissart table of the Tau solution $y_{150}$ with $\text{\rm tol}=10^{-5}$, $1\leq p,q\leq 25$.
  • Figure 4: Absolute error in Tau solution $y_{150}$, in black. Absolute error in the filter $\Phi_{10,10}(y_{150})$, in blue.
  • Figure 5: $w$-norm of the functions errors $e_{n}$, $n=50, \ 151, \ 501$ and $1000$, for values of $\alpha\in ]0,1[$.
  • ...and 2 more figures

Theorems & Definitions (10)

  • Example 1
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Corollary 4
  • Corollary 5
  • Corollary 6
  • Example 2