Uniformly accurate multiscale time integrators for highly oscillatory second order differential equations

TL;DR

This paper introduces two multiscale time integrators for highly oscillatory second order differential equations, achieving uniform accuracy and improved meshing strategies compared to classical methods.

Contribution

The paper proposes and analyzes two novel multiscale time integrators based on frequency and amplitude decomposition, with rigorous error bounds and enhanced efficiency for small epsilon regimes.

Findings

Error bounds of O(τ²/ε²) and O(ε²) established for the two MTIs.

Uniform convergence with linear and quadratic rates depending on ε and τ.

Numerical tests confirm the theoretical error bounds and improved meshing strategies.

Abstract

In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $0<\varepsilon\le1$. In fact, the solution to this equation propagates waves with wavelength at $O(\varepsilon^2)$ when $0<\varepsilon\ll 1$, which brings significantly numerical burdens in practical computation. We rigorously establish two independent error bounds for the two MTIs at $O(\tau^2/\varepsilon^2)$ and $O(\varepsilon^2)$ for $\varepsilon\in(0,1]$ with $\tau>0$ as step size, which imply that the two MTIs converge uniformly with linear convergence rate at $O(\tau)$ for $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\le \tau$. Thus the meshing strategy requirement (or $\varepsilon$-scalability) of the two MTIs is $\tau=O(1)$ for $0<\varepsilon\ll 1$, which is significantly improved from $\tau=O(\varepsilon^3)$ and $\tau=O(\varepsilon^2)$ requested by finite difference methods and exponential wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards better understanding on the convergence and resolution properties of the two MTIs. In addition, numerical results support the two error bounds very well.