2025/11/19 by Qi, Zhihao, Deng, Weibing, Zhu, Fuhai · 1 citation
Mathematics · Computer Science · #Numerical methods for differential equations #Matrix Theory and Algorithms #Advanced Mathematical Modeling in Engineering
paper · doi:10.48550/arxiv.2511.15104
This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter ε, with purely imaginary eigenvalues, and an ε-independent nonlinear part. When 0<ε≪ 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(hk+1). In particular, an improved order of O(ε hk) is attained when h is larger than the scale of ε. These theoretical findings are further applied to second-order oscillatory systems, leading to improved uniform accuracy with respect to ε. Finally, numerical experiments confirm the optimality of the derived error estimates.