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：本文的主要目的是针对Benjamin-Ono方程设计一种局部间断Galerkin方法。我们分析了半离散LDG格式的稳定性与误差估计，证明了该格式对于一般非线性通量是$L^2$稳定的，且收敛阶为$\mathcal{O}(h^{k+1/2})$。此外，我们采用四阶四级Runge-Kutta方法构造了全离散LDG格式，并在适当的时间步长约束下，利用两步和三步稳定性方法确保所设计的格式在线性通量情形下具有强稳定性。最后通过数值算例验证了该方法的有效性与精确性。