Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Numerical integrator for highly oscillatory differential equations based on the Neumann series

Rafał Perczyński, Grzegorz Madejski

TL;DR

The paper develops a 4th-order local time-stepping scheme for linear, highly oscillatory PDEs by combining a Neumann-series representation with Filon quadrature. By truncating the Neumann expansion to the first four terms and approximating the resulting oscillatory integrals with cubic Hermite Filon quadrature, the method achieves accuracy that can improve with increasing oscillation parameter $\omega$, while remaining stable for small $h$ and enabling easy higher-order refinement. The approach is applicable to equations with elliptic operators, including the heat and wave equations, and is shown to outperform Magnus-based exponential integrators in highly oscillatory regimes. Numerical experiments in 1D and 2D corroborate the theoretical error bounds and illustrate robustness to large $\omega$, with explicit treatment of resonance and nonresonance frequency components. Overall, the work provides a versatile, spectrally friendly integrator that extends oscillatory solvers beyond traditional step-size restrictions.

Abstract

We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.

Numerical integrator for highly oscillatory differential equations based on the Neumann series

TL;DR

The paper develops a 4th-order local time-stepping scheme for linear, highly oscillatory PDEs by combining a Neumann-series representation with Filon quadrature. By truncating the Neumann expansion to the first four terms and approximating the resulting oscillatory integrals with cubic Hermite Filon quadrature, the method achieves accuracy that can improve with increasing oscillation parameter , while remaining stable for small and enabling easy higher-order refinement. The approach is applicable to equations with elliptic operators, including the heat and wave equations, and is shown to outperform Magnus-based exponential integrators in highly oscillatory regimes. Numerical experiments in 1D and 2D corroborate the theoretical error bounds and illustrate robustness to large , with explicit treatment of resonance and nonresonance frequency components. Overall, the work provides a versatile, spectrally friendly integrator that extends oscillatory solvers beyond traditional step-size restrictions.

Abstract

We propose a third-order numerical integrator based on the Neumann series and the Filon quadrature, designed mainly for highly oscillatory partial differential equations. The method can be applied to equations that exhibit small or moderate oscillations; however, counter-intuitively, large oscillations increase the accuracy of the scheme. With the proposed approach, the convergence order of the method can be easily improved. Error analysis of the method is also performed. We consider linear evolution equations involving first- and second-time derivatives that feature elliptic differential operators, such as the heat equation or the wave equation. Numerical experiments consider the case in which the space dimension is greater than one and confirm the theoretical study.
Paper Structure (10 sections, 6 theorems, 99 equations, 6 figures)

This paper contains 10 sections, 6 theorems, 99 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Derivation of the method
  4. Local error analysis
  5. Positive frequencies
  6. The case involving negative frequencies
  7. Application of the method to the wave equation
  8. Numerical examples
  9. Comparison with other methods
  10. Conclusion

Key Result

Lemma 1

Let $F(\tau)$ be a 4 times continuously differentiable, vector-valued function, and let $p(\tau)$ be a cubic Hermite interpolation polynomial such that $p(0)=F(0)$, $p(h)=F(h)$, $p'(0)=F'(0)$, $p'(h)=F'(h)$. Then the error of the Filon method satisfies

Figures (6)

  • Figure 1: Numerical approximation of the solution to equation (\ref{['heat_example_t']}). Error versus time step (left graph) and error versus parameter $\omega$ (right graph).
  • Figure 2: Numerical approximation of the solution to equation (\ref{['2dheat_example']}). Error versus time step (left graph) and error versus parameter $\omega$ (right graph).
  • Figure 3: Numerical approximation of the solution to equation (\ref{['wave_example_bz']}). Error versus time step (left graph) and error versus $\omega$ (right graph).
  • Figure 4: Numerical approximation of the solution to equation (\ref{['wave_example']}). Error versus time step (left figure) and error versus $\omega$ (right figure).
  • Figure 5: The error of approximating the solutions of equations (\ref{['heat_example_t']}), (\ref{['2dheat_example']}), (\ref{['wave_example_bz']}), and (\ref{['wave_example']}), for $\omega$ ranging from 5 to 1000, with step sizes $h=1$.
  • ...and 1 more figures

Theorems & Definitions (10)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • Theorem 1
  • Theorem 2
  • proof