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Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations

Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations
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Year:    2014

Author:    Weizhu Bao, Xuanchun Dong, Xiaofei Zhao

Journal of Mathematical Study, Vol. 47 (2014), Iss. 2 : pp. 111–150

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 $0 < \varepsilon≤ 1.$ In fact, the solution to this equation propagates waves with wavelength at $O(\varepsilon^2)$ when $0<\varepsilon≪1,$ 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 $\varepsilon ∈ (0,1]$ with $\tau > 0$ as step size, which imply that the two MTIs converge uniformly with linear convergence rate at $O(\tau)$ for $ε ∈ (0,1]$ and optimally with quadratic convergence rate at $O(\tau^2)$ in the regimes when either $ε=O(1)$ or $0<ε≤\tau.$ Thus the meshing strategy requirement (or $ε$-scalability) of the two MTIs is $\tau =O(1)$ for $0<ε≪1,$ which is significantly improved from $\tau =O(ε^3)$ and $\tau =O(ε^2)$ 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jms.v47n2.14.01

Journal of Mathematical Study, Vol. 47 (2014), Iss. 2 : pp. 111–150

Published online:    2014-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    40

Keywords:    Highly oscillatory differential equations multiscale time integrator uniformly accurate multiscale decomposition exponential wave integrator.

Author Details

Weizhu Bao

Xuanchun Dong

Xiaofei Zhao

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