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

TL;DR

This paper provides a rigorous error analysis of a new class of explicit exponential integrators for highly oscillatory ODEs, demonstrating their high-order convergence and improved accuracy under certain conditions.

Contribution

It introduces a novel error analysis for exponential integrators with local linear extension, showing uniform convergence and improved order for highly oscillatory problems.

Findings

Achieves uniform convergence order of O(h^{k+1}) with polynomial degree k

Attains order O(ε h^k) when step size exceeds ε scale

Numerical experiments confirm theoretical error estimates

Abstract

This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an $\varepsilon$-independent nonlinear part. When $0<\varepsilon\ll 1$, the rapidly oscillatory nature of the solution imposes severe constraints on step size selection and numerical accuracy, leading to considerable computational difficulties. Inspired by a linearization technique that introduces auxiliary polynomial variables, a new family of explicit exponential integrators has recently been proposed. These methods do not require the linear part to be diagonal or to have eigenvalues that are integer multiples of a fixed value - a common assumption in multiscale approaches - and they achieve arbitrarily high orders of convergence without imposing order conditions. The main contribution of this work is to provide a rigorous error analysis for this new class of methods under a bounded oscillatory energy condition. To this end, we first establish the equivalence between the high-dimensional system and the original problem using algebraic techniques. Building on these foundational results, we prove that the numerical schemes, when employing auxiliary polynomial variables of degree $k$, achieve a uniform convergence order of $O(h^{k+1})$. In particular, an improved order of $O(\varepsilon h^k)$ is attained when $h$ is larger than the scale of $\varepsilon$. These theoretical findings are further applied to second-order oscillatory systems, leading to improved uniform accuracy with respect to $\varepsilon$. Finally, numerical experiments confirm the optimality of the derived error estimates.