Stability of fully Discrete Local Discontinuous Galerkin method for the generalized Benjamin-Ono equation

TL;DR

This paper develops and analyzes fully discrete local discontinuous Galerkin schemes for the generalized Benjamin-Ono equation, proving stability and convergence, and validating with numerical examples.

Contribution

It introduces a fully discrete LDG scheme using Crank-Nicolson and Runge-Kutta methods, proving stability and higher-order convergence for the generalized Benjamin-Ono equation.

Findings

Proved $L^2$-stability of semi-discrete LDG scheme.

Established stability of the fully discrete CN-LDG scheme.

Validated optimal order of accuracy through numerical examples.

Abstract

The main purpose of this paper is to design a fully discrete local discontinuous Galerkin (LDG) scheme for the generalized Benjamin-Ono equation. First, we proved the $L^2$-stability for the proposed semi-discrete LDG scheme and obtained a sub-optimal order of convergence for general nonlinear flux. We develop a fully discrete LDG scheme using the Crank-Nicolson (CN) method and fourth-order fourth-stage Runge-Kutta (RK) method in time. Adapting the methodology established for the semi-discrete scheme, we demonstrate the stability of the fully discrete CN-LDG scheme for general nonlinear flux. Additionally, we consider the fourth-order RK-LDG scheme for higher order convergence in time and prove that it is strongly stable under an appropriate time step constraint by establishing a \emph{three-step strong stability} estimate for linear flux. Numerical examples associated with soliton solutions are provided to validate the efficiency and optimal order of accuracy for both methods.