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arXiv · 2606.13807

A WKB-related time-stepping scheme for differential equations describing oscillatory systems

Juliane Rosemeier·Rupert Klein
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Abstract

In this study, we present a novel time-stepping scheme for multiscale differential equations describing oscillatory systems with well-separated scales, where the scale separation is controlled by a small parameter $\epsilon$. The time-stepping method is related to a multi-modal WKB approximation and relies on a transformation of variables derived in this work. The analysis reveals that, in the transformed formulation, the leading-order oscillations are either eliminated or appear only at higher asymptotic order. The method is applied to ordinary differential equations, including the well-known van der Pol oscillator. We investigate the accuracy of the proposed method numerically for different parameter regimes, in particular for decreasing values of $\epsilon$, and study how the parameters of the numerical scheme must be adapted as $\epsilon$ is reduced. In the presented numerical tests, the computational cost remains bounded as $\epsilon$ is decreased.

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Juliane Rosemeier, Rupert Klein. 2026-06-11. A WKB-related time-stepping scheme for differential equations describing oscillatory systems. https://arxiv.org/abs/2606.13807

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