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Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential

Rafał Perczyński, Antoni Augustynowicz

TL;DR

This work develops an analytic Modulated Fourier Expansion for linear PDEs with multifrequency highly oscillatory potentials by representing the solution as a convergent Neumann series in the Sobolev space $H^{2p}(\Omega)$. It derives a fully analytic computation of the expansion coefficients and a precise asymptotic expansion for the highly oscillatory multivariate integrals underlying each Neumann-term, under a nonresonance condition, and extends the framework to second-order time derivatives (wave/Klein–Gordon) equations. The authors provide error bounds of the form $\mathcal{O}(\omega^{-(r+1)})$ for a truncated expansion with parameter $r$, and demonstrate the approach via numerical experiments showing accuracy for practical $\omega$ and systems. The results enable long-time, high-frequency simulations without solving large differential systems, and point to extensions including Filon-type quadrature and more general oscillatory potentials.

Abstract

Highly oscillatory differential equations present significant challenges in numerical treatments. The Modulated Fourier Expansion (MFE), used as an ansatz, is a commonly employed tool as a numerical approximation method. In this article, the Modulated Fourier Expansion is analytically derived for a linear partial differential equation with a multifrequency highly oscillatory potential. The solution of the equation is expressed as a convergent Neumann series within the appropriate Sobolev space. The proposed approach enables, firstly, to derive a general formula for the error associated with the approximation of the solution by MFE, and secondly, to determine the coefficients for this expansion -- without the need to solve numerically the system of differential equations to find the coefficients of MFE. Numerical experiments illustrate the theoretical investigations.

Asymptotic expansions for the solution of a linear PDE with a multifrequency highly oscillatory potential

TL;DR

This work develops an analytic Modulated Fourier Expansion for linear PDEs with multifrequency highly oscillatory potentials by representing the solution as a convergent Neumann series in the Sobolev space . It derives a fully analytic computation of the expansion coefficients and a precise asymptotic expansion for the highly oscillatory multivariate integrals underlying each Neumann-term, under a nonresonance condition, and extends the framework to second-order time derivatives (wave/Klein–Gordon) equations. The authors provide error bounds of the form for a truncated expansion with parameter , and demonstrate the approach via numerical experiments showing accuracy for practical and systems. The results enable long-time, high-frequency simulations without solving large differential systems, and point to extensions including Filon-type quadrature and more general oscillatory potentials.

Abstract

Highly oscillatory differential equations present significant challenges in numerical treatments. The Modulated Fourier Expansion (MFE), used as an ansatz, is a commonly employed tool as a numerical approximation method. In this article, the Modulated Fourier Expansion is analytically derived for a linear partial differential equation with a multifrequency highly oscillatory potential. The solution of the equation is expressed as a convergent Neumann series within the appropriate Sobolev space. The proposed approach enables, firstly, to derive a general formula for the error associated with the approximation of the solution by MFE, and secondly, to determine the coefficients for this expansion -- without the need to solve numerically the system of differential equations to find the coefficients of MFE. Numerical experiments illustrate the theoretical investigations.
Paper Structure (9 sections, 11 theorems, 119 equations, 1 figure, 2 tables)

This paper contains 9 sections, 11 theorems, 119 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Representation of the solution as the Neumann series
  3. Presenting terms of the Neumann series as a sum of multivariate integrals
  4. Asymptotic expansion of a highly oscillatory integral subject to nonresonance condition
  5. Error analysis. First four terms of the Modulated Fourier expansion
  6. Asymptotic expansion -- integrals with resonance points
  7. A highly oscillatory wave equation
  8. Numerical examples
  9. Concluding remarks

Key Result

Lemma 1

Suppose that Assumption assumption1 is satisfied. Let $u \in V$ ($V$ is the domain of operator $T$ defined in (T)) and $2p>m/2$. Then there exist a constant $M$ (depending only on $p$ and $m$), such that where $C_1:= \max_{t \in [0,t^\star]}\|\mathrm{e}^{t\mathcal{L}}\|_{D(\mathcal{L})\leftarrow D(\mathcal{L})}$, $C_2:= \max_{t \in [0,t^\star]}\|f(t)\|$ and $C_3:= \max_{t \in [0,t^\star]}\|u(t)\|

Figures (1)

  • Figure 1: $L^2$ norm of the error of the method for the equation (\ref{['heat_example']}) for $t=3$. We use a base--10 log scale.

Theorems & Definitions (31)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Remark 1
  • Lemma 2
  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • ...and 21 more