Harmonic-Enriched Reproducing Kernel Approximation for Highly Oscillatory Differential Equations
Publication: Journal of Engineering Mechanics
Volume 146, Issue 4
Abstract
The harmonic-enriched reproducing kernel (HRK) approximation together with collocation method is introduced to circumvent the discretization restriction for highly oscillatory partial differential equations (PDEs). It is first shown that to embed the harmonic function with a desired frequency in the HRK, both sine and cosine with the same frequency should be included in the basis vector for construction of HRK approximation. The HRK and its implicit derivatives are then used in the collocation method to effectively obtain solutions of oscillatory PDEs. The standard monomials can be included together with harmonic functions in the HRK and the reproducing conditions can be exactly satisfied with a complete set of basis functions. For PDEs with semi-harmonic solutions, the present method yields more accurate results compared with the standard reproducing kernel (RK) when a coarse discretization is used. On the other hand, when the discretization is refined, the HRK exhibits a similar convergence behavior as the standard RK. The effectiveness of the present method is demonstrated using highly oscillatory 2nd order and 4th order PDEs. The accuracy and performance of this method are compared with standard RK with the collocation method and the finite element method (FEM).
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Data Availability Statement
Some or all data, models, or code generated or used during the study are available from the corresponding author by request [codes for implementation of the proposed harmonic RKCM (H-RKCM) to one-dimensional, two-dimensional, and second- and fourth-order problems solved in this paper].
Acknowledgments
Research reported in this paper was supported by National Science Foundation (NSF) under Award No. 1463501.
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Published In
Journal of Engineering Mechanics
Volume 146 • Issue 4 • April 2020
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©2020 American Society of Civil Engineers.
History
Received: Oct 23, 2018
Accepted: Jul 30, 2019
Published online: Feb 3, 2020
Published in print: Apr 1, 2020
Discussion open until: Jul 3, 2020
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