Zhihao Qi | Searcharxiv

arXiv2026

High-Order Exponential Integrators with Improved Uniform Accuracy for Charged-Particle in a Perpendicular Strong Magnetic Field

This paper considers a class of charged-particle dynamics problems in which the particle is subjected to a magnetic force, with a magnetic flux density inversely proportional to a small parameter $0<\varepsilon\ll 1$, and a nonlinear electric force. The resulting highly oscillatory behavior poses significant challenges for numerical computation. To enhance the performance of exponential integrators (EIs), this paper employs a technique that linearizes the ordinary differential equation through a dimension-raising approach. Based on this technique, a new family of EIs is developed that achieves arbitrarily high order. For short-time simulations on the interval $[0,T]$, it is rigorously proved that the proposed method--which employs auxiliary polynomials of degree $k$ and a time step $Δt$--satisfies two distinct error bounds: $O(\varepsilon Δt^{k+1})$ and $O(\varepsilon^{k+2})$. The latter bound guarantees that the algorithm stays accurate even when the step size is of order $O(1)$. Furthermore, when a large step size $\varepsilon^{-1}Δt$ is used to simulate the long-term dynamics over $[0,\varepsilon^{-1}T]$, the numerical scheme attains a uniform convergence rate of $O(Δt^{k+1})$. Several numerical experiments confirm these theoretical results.

math.NA↗

arXiv2026

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

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.

math.NA↗