The main purpose of this paper is to design a local discontinuous Galerkin (LDG) method for the Benjamin-Ono equation. We analyze the stability and error estimates for the semi-discrete LDG scheme. We prove that the scheme is $L^2$-stable and it converges at a rate $\mathcal{O}(h^{k+1/2})$ for general nonlinear flux. Furthermore, we develop a fully discrete LDG scheme using the four-stage fourth order Runge-Kutta method and ensure the devised scheme is strongly stable in case of linear flux using two-step and three-step stability approach under an appropriate time step constraint. Numerical examples are provided to validate the efficiency and accuracy of the method.

翻译：本文的主要目的是为 Benjamin-Ono 方程设计一种局部间断 Galerkin (LDG) 方法。我们分析了半离散 LDG 格式的稳定性和误差估计。我们证明了该格式是 $L^2$ 稳定的，并且对于一般非线性通量，其收敛速度为 $\mathcal{O}(h^{k+1/2})$。此外，我们利用四阶四阶段 Runge-Kutta 方法开发了一种全离散 LDG 格式，并在适当的步长约束下，通过两步和三步稳定性分析方法，确保了所设计格式在线性通量情况下的强稳定性。数值算例验证了该方法的效率和精度。