An efficient spectral method for two-dimensional Benjamin–Bona–Mahony–Burgers equation

Purpose In this paper, we propose an efficient spectral method for solving the two-dimensional Benjamin–Bona–Mahony–Burgers equation. The new basis functions align well with the problem, the discrete system is sparse and can be efficiently inverted, and the numerical solutions exhibit spectral accuracy in space. Design/methodology/approach To efficiently simulate the two-dimensional Benjamin–Bona–Mahony–Burgers equation, we utilize transformed generalized Jacobi polynomials and construct the basis functions using the tensor product of these newly introduced polynomials. We provide relevant approximation results. Subsequently, we propose a spectral scheme for the underlying problem, and prove the well-posedness of the scheme, along with the boundedness and energy dissipation of the numerical solutions. We analyze the generalized stability and convergence of the numerical solution of the proposed scheme. Some numerical simulations are presented to demonstrate the efficacy of this newly proposed method. Findings The new basis functions generated by tensor product of the transformed Jacobi polynomial align well with the underlying problem and simplify the theoretical analysis. The spatial discrete system is sparse and can be efficiently inverted. The numerical solutions exhibit spectral accuracy in space. Originality/value We introduce transformed generalized Jacobi polynomials to construct basis functions and present relevant approximation results. We propose an efficient spectral scheme for the two-dimensional Benjamin–Bona–Mahony–Burgers equation, accompanied by optimal error analysis. This new approach achieves spectral accuracy. Moreover, the proposed method and the techniques developed in this work can be applied to simulate a wide range of other nonlinear problems.