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

Mathematics

Tony Liu¹  Benjamin Akers¹  Jonah Reeger² 

[1] Department of Mathematics and Statistics, Air Force Institute of Technology, Dayton, OH 45433, USA;Independent Researcher, Bellbrook, OH 45305, USA;

关键词: radial basis functions;    finite difference methods;    traveling waves;    non-uniform grids;   

DOI  :  10.3390/math9010065

来源: DOAJ

【 摘 要 】

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 $\mathbb{R}$, 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.

【 授权许可】

Unknown