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Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations by Bao, Weizhu; Dong, Xuanchun; Zhao, Xiaofei is a scholarly article available to read on EtoBox.

What is Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations about?

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 inte

Author

Bao, Weizhu; Dong, Xuanchun; Zhao, Xiaofei

Published

2012

Language

EN