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

Benjamin Akers,Jonah Reeger,Tony Liu

doi:10.3390/math9010065

Benjamin Akers, Jonah Reeger + Show 1 more

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https://doi.org/10.3390/math9010065

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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.

Highlights

In this paper we use the benjamin–ono equation as a test-bed for new radial basis function-finite differencing (RBF-FD) simulations of nonlocal wave equations on nonuniform grids
In this paper we look at an extension of radial basis functions (RBFs)-FD to pseudo-differential operators
In the process of simulating the Benjamin–Ono equation, we present a simple framework for using RBF-FD to approximate pseudo-differential operators

Summary

Introduction

In this paper we use the benjamin–ono equation as a test-bed for new radial basis function-finite differencing (RBF-FD) simulations of nonlocal wave equations on nonuniform grids. The Benjamin–Ono equation presents the numerical challenges of numerical stiffness, a nonlocal pseudo-differential operator, and localized traveling solutions with slow decay. On R, Equation (1) supports traveling solitary waves solutions: Received: 24 November 2020 Accepted: 18 December 2020 Published: 30 December 2020. The Benjamin–Ono equation is known to be well-posed [1] and integrable. It can be solved with inverse scattering, and many exact solution profiles are known [2,3]. It has been numerically simulated many times, both in the periodic setting [4] and on R [5,6]

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