Convergence Rates of a Fully Discrete Galerkin Scheme for the Benjamin–Ono Equation

Abstract

We consider a recently proposed fully discrete Galerkin scheme for the Benjamin-Ono equation which has been found to be locally convergent in finite time for initial data in L 2 (R). By assuming that the initial data is sufficiently regular we obtain theoretical convergence rates for the scheme both in the full line and periodic versions of the associated initial value problem. These rates are illustrated with some numerical examples.

Stability and Convergence analysis of a Crank–Nicolson Galerkin scheme for the fractional Korteweg-de Vries equation

“…In a similar manner, for α = 1, (1.1) has been investigated numerically by several authors. A convergent finite difference scheme is developed in [6] and Galtung [11,13] designed a fully discrete Galerkin scheme for the BO equation. However, there is a limited literature concerning the numerical framework specifically for the fractional KdV equation (1.1) with α ∈ (1, 2).…”

A convergent Crank-Nicolson Galerkin scheme for the Benjamin-Ono equation

“…due to Sobolev embedding. In the last line we apply (24) to the sum, and the inverse equality (14) combined with the assumption ∆t ≤ C∆x 2 to the second term to conclude that (38) holds. For the third relation, note that…”

“…It should also be pointed out that this is a complicated example, as one has to approximate the nonlinear interaction between two passing solitons. Finally, we refer to [14] for a study of theoretical convergence rates for this scheme and a modified scheme aimed at the periodic version of (1), given sufficiently regular initial data.…”

“…We note that the results presented in the current paper is in full based on [15]. For a treatment of convergence rates for the scheme discussed here given sufficiently regular initial data the reader is referred to [14].…”

“…Estimating the term involving the Hilbert transform as before, using Young's inequality and (14) leads to…”