Stability of a Fully Discrete Local Discontinuous Galerkin Method for the Generalized Benjamin–Ono Equation

ABSTRACT 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 prove the ‐stability for the proposed semi‐discrete LDG scheme and obtained a suboptimal order of convergence for power 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, to the CN‐LDG scheme, we establish ‐stability and error estimates for general power nonlinear flux. Additionally, for the linearized equation corresponding to zero or linear flux, we consider the fourth‐order RK‐LDG scheme for higher‐order convergence in time. We demonstrate that it is strongly stable under the step‐size condition by establishing a three‐step strong stability estimate and subsequently, the error estimates are obtained for this case. Numerical examples associated with soliton solutions are provided to validate the efficiency and expected order of accuracy for both methods.