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Error Analysis on a Novel Class of Exponential Integrators with Local Linear Extension Techniques for Highly Oscillatory ODEs

Zhihao Qi, Weibing Deng, Fuhai Zhu

TL;DR

This work tackles highly oscillatory ODEs with a fast linear part $\frac{1}{\varepsilon}A$ by introducing local linear extension exponential integrators that raise the system dimension with polynomial extension variables. Through a rigorous algebraic framework, it is shown that the extension retains the spectral properties of the original linear part and permits explicit, high-order integration without order conditions. The main results establish uniform convergence: $O(h^{k+1})$ for small steps $h<\varepsilon$ and $O(\varepsilon h^{k})$ for large steps $h>\varepsilon$ under bounded oscillatory energy, with extensions to second-order equations via adiabatic transformations and invariant subspaces. Numerical experiments confirm the optimality of these error bounds and demonstrate robust, $\varepsilon$-uniform accuracy across regimes, outperforming some traditional exponential integrators in highly oscillatory settings.

Abstract

This paper studies a class of non-autonomous highly oscillatory ordinary differential equations (ODEs) featuring a linear component inversely proportional to a small parameter $\varepsilon$ with purely imaginary eigenvalues, alongside an $\varepsilon$-independent nonlinear component. When $0<\varepsilon\ll 1$, the rapidly oscillatory solution constrains the step size selection and numerical accuracy, resulting in significant computational challenges. Motivated by linearization through introducing auxiliary polynomial variables, a new class of explicit exponential integrators (EIs) has recently been developed. The methods do not require the linear part to be diagonal or with all eigenvalues to be integer multiples of a fixed value - a general assumption in multiscale methods - and attain arbitrarily high convergence order without any order conditions. The main contribution of this work is to establish a rigorous error analysis for the new class of methods. To do this, we first demonstrate the equivalence between the high-dimensional system and the original problem by employing algebraic techniques. Building upon these fundamental results, we prove that the numerical schemes have a uniform convergence order of $O(h^{k+1})$ for the solution when using at most $k$-degree auxiliary polynomial variables with time step sizes smaller than $\varepsilon$. For larger step sizes under the bounded oscillatory energy condition, the methods achieve a convergence order of $O(\varepsilon h^k)$ for the solution. These theoretical results are further applied to second-order oscillatory equations, yielding improved uniform accuracy with respect to $\varepsilon$. Finally, numerical experiments confirm the optimality of the derived error estimates.

Error Analysis on a Novel Class of Exponential Integrators with Local Linear Extension Techniques for Highly Oscillatory ODEs

TL;DR

This work tackles highly oscillatory ODEs with a fast linear part by introducing local linear extension exponential integrators that raise the system dimension with polynomial extension variables. Through a rigorous algebraic framework, it is shown that the extension retains the spectral properties of the original linear part and permits explicit, high-order integration without order conditions. The main results establish uniform convergence: for small steps and for large steps under bounded oscillatory energy, with extensions to second-order equations via adiabatic transformations and invariant subspaces. Numerical experiments confirm the optimality of these error bounds and demonstrate robust, -uniform accuracy across regimes, outperforming some traditional exponential integrators in highly oscillatory settings.

Abstract

This paper studies a class of non-autonomous highly oscillatory ordinary differential equations (ODEs) featuring a linear component inversely proportional to a small parameter with purely imaginary eigenvalues, alongside an -independent nonlinear component. When , the rapidly oscillatory solution constrains the step size selection and numerical accuracy, resulting in significant computational challenges. Motivated by linearization through introducing auxiliary polynomial variables, a new class of explicit exponential integrators (EIs) has recently been developed. The methods do not require the linear part to be diagonal or with all eigenvalues to be integer multiples of a fixed value - a general assumption in multiscale methods - and attain arbitrarily high convergence order without any order conditions. The main contribution of this work is to establish a rigorous error analysis for the new class of methods. To do this, we first demonstrate the equivalence between the high-dimensional system and the original problem by employing algebraic techniques. Building upon these fundamental results, we prove that the numerical schemes have a uniform convergence order of for the solution when using at most -degree auxiliary polynomial variables with time step sizes smaller than . For larger step sizes under the bounded oscillatory energy condition, the methods achieve a convergence order of for the solution. These theoretical results are further applied to second-order oscillatory equations, yielding improved uniform accuracy with respect to . Finally, numerical experiments confirm the optimality of the derived error estimates.

Paper Structure

This paper contains 11 sections, 15 theorems, 108 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The local linear extension exponential integrators
  3. Algebraic properties of $A_1^{[k]}$
  4. Linear extension lemma
  5. Invariant subspaces and block lower triangular diagonalization
  6. Convergence analysis
  7. Convergence results when $h<\varepsilon$
  8. Convergence results with bounded oscillatory energy when $h>\varepsilon$
  9. Application on second-order equations
  10. Numerical results
  11. Conclusion

Key Result

Lemma 3.1

Suppose the local linear extension operation acting on (P) gives $A_1^{[k]}(\hat{\mathbf{x}})$ with $A_1$ defined in (P02). Then $A_1^{[k]}(\hat{\mathbf{x}})$ is diagonalizable with all eigenvalues having zero real parts.

Figures (4)

  • Figure 1: Error versus step size $h$. The first row corresponds to the weakly oscillatory case ($\varepsilon=1/2^2$), while the second row represents the highly oscillatory regime ($\varepsilon=1/2^8$). The left and right columns display the numerical errors for $\mathbf{y}$ and $\dot{\mathbf{y}}$, respectively.
  • Figure 2: Error versus step size $\varepsilon$. The first row corresponds to the small step size case ($h=1/2^6$), while the second row represents the large step size case ($h=1/2$). The left and right columns display the numerical errors for $\mathbf{y}$ and $\dot{\mathbf{y}}$, respectively.
  • Figure 3: $error(\mathbf{u})$ versus step size $h$. The first row corresponds to the weakly oscillatory case ($\varepsilon=1/2^2$), while the second row represents the highly oscillatory regime ($\varepsilon=1/2^8$). The left, middle, and right columns correspond to second-, third-, and fourth-order methods, respectively.
  • Figure 4: $error(\mathbf{u})$ versus $\varepsilon$. The first row corresponds to the large step size case ($h=1/2^3$), while the second row represents the small step size case ($h=1/2^8$). The left, middle, and right columns correspond to second-, third-, and fourth-order methods, respectively.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 19 more