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.

翻译：本文的主要目的是为广义Benjamin-Ono方程设计一种全离散局部间断Galerkin（LDG）格式。首先，我们分析了半离散LDG格式的稳定性，并证明了该格式对于一般非线性通量是$L^2$稳定的。我们采用Crank-Nicolson（CN）方法和四阶四级Runge-Kutta（RK）方法进行时间离散，构建了全离散LDG格式。通过沿用为半离散格式建立的方法论，我们证明了全离散CN-LDG格式对于一般非线性通量的稳定性。此外，我们考虑了具有更高时间收敛阶的四阶RK-LDG格式，并通过为线性通量建立\emph{三步强稳定性}估计，证明了其在适当时间步长约束下是强稳定的。数值算例验证了两种方法的效率与最优收敛阶。