A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation

Authors

Benjamin F. Akers, Air Force Institute of TechnologyFollow
Tony Liu, Air Force Institute of TechnologyFollow
Jonah A. Reeger

Document Type

Article

Publication Date

12-30-2020

Abstract

A radial basis function-finite differencing (RBF-FD) scheme was applied to the initial value problem of the Benjamin–Ono equation. The Benjamin–Ono equation has traveling wave solutions with algebraic decay and a nonlocal pseudo-differential operator, the Hilbert transform. When posed on ℝ, the former makes Fourier collocation a poor discretization choice; the latter is challenging for any local method. We develop an RBF-FD approximation of the Hilbert transform, and discuss the challenges of implementing this and other pseudo-differential operators on unstructured grids. Numerical examples, simulation costs, convergence rates, and generalizations of this method are all discussed.

Comments

© The Authors. Licensee MDPI, Basel, Switzerland.

Published by MDPI online in December 2020 ahead of inclusion in the January 2021 issue as cited.

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. CC BY 4.0

DOI

10.3390/math9010065

Source Publication

Mathematics (eISSN 2227-7390)

Recommended Citation

Akers B, Liu T, Reeger J. A Radial Basis Function Finite Difference Scheme for the Benjamin–Ono Equation. Mathematics. 2021; 9(1):65. https://doi.org/10.3390/math9010065