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。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，我们利用四阶四级Runge-Kutta方法发展了一种全离散LDG格式，并采用两步和三步稳定性分析方法，在适当的时间步长约束下，确保了所设计格式在线性通量情形下的强稳定性。数值算例验证了该方法的效率和精度。