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

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 analyze the stability for the semi-discrete LDG scheme and we prove that the scheme is $L^2$-stable 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 are provided to validate the efficiency and optimal order of accuracy for both the methods.