Volume 47, Issue 2
Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations

J. Math. Study, 47 (2014), pp. 111-150.

Published online: 2014-06

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• 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.

• Keywords

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

65L05, 65L20, 65L70

matbaowz@nus.edu.sg (Weizhu Bao)

dong.xuanchun@gmail.com (Xuanchun Dong)

zhxfnus@gmail.com (Xiaofei Zhao)

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@Article{JMS-47-111, author = {Weizhu and Bao and matbaowz@nus.edu.sg and 13253 and Department of Mathematics and Center for Computational Science and Engineering, National University of Singapore, Singapore 119076, Singapore and Weizhu Bao and Xuanchun and Dong and dong.xuanchun@gmail.com and 11428 and Beijing Computational Science Research Center, Beijing 100084, P.R. China and Xuanchun Dong and Xiaofei and Zhao and zhxfnus@gmail.com and 11328 and Department of Mathematics, National University of Singapore, Singapore 119076, Singapore and Xiaofei Zhao}, title = {Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations}, journal = {Journal of Mathematical Study}, year = {2014}, volume = {47}, number = {2}, pages = {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.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v47n2.14.01}, url = {http://global-sci.org/intro/article_detail/jms/9951.html} }
TY - JOUR T1 - Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations AU - Bao , Weizhu AU - Dong , Xuanchun AU - Zhao , Xiaofei JO - Journal of Mathematical Study VL - 2 SP - 111 EP - 150 PY - 2014 DA - 2014/06 SN - 47 DO - http://doi.org/10.4208/jms.v47n2.14.01 UR - https://global-sci.org/intro/article_detail/jms/9951.html KW - Highly oscillatory differential equations, multiscale time integrator, uniformly accurate, multiscale decomposition, exponential wave integrator. AB -

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.

Weizhu Bao, Xuanchun Dong & Xiaofei Zhao. (2019). Uniformly Accurate Multiscale Time Integrators for Highly Oscillatory Second Order Differential Equations. Journal of Mathematical Study. 47 (2). 111-150. doi:10.4208/jms.v47n2.14.01
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