Efficient spectral-Galerkin method for the “Good” Boussinesq equation using generalized Jacobi polynomials

Purpose The purpose of this paper is to develop an efficient spectral-Galerkin method for solving the “Good” Boussinesq (GB) equation with homogeneous boundary conditions. The study aims to overcome the computational challenges associated with unbounded domains while preserving high-order accuracy and computational efficiency. Design/methodology/approach A spectral-Galerkin framework based on generalized Jacobi polynomials (GJPs) is proposed for the spatial discretization of the GB equation. Exploiting the localized structure of solitary-wave solutions, the original problem posed on an unbounded domain is truncated to a finite interval. The resulting semi-discrete system is advanced in time using an explicit fourth-order Runge–Kutta scheme. Theoretical properties of the method, including boundedness, generalized stability, and convergence, are established through an energy-based analysis. Detailed implementation procedures are provided to facilitate practical computation. Findings The proposed method yields a sparse algebraic system due to the use of GJP basis functions, thereby significantly reducing computational cost and memory requirements. The selected GJP basis functions inherently satisfy the homogeneous Neumann boundary conditions, simplifying both the formulation and theoretical analysis. Rigorous analysis confirms the boundedness, generalized stability and convergence of the scheme. Numerical results confirm the theoretical findings and demonstrate spectral accuracy in space. Originality/value The proposed method combines domain truncation and GJP-based spectral approximation to efficiently solve the GB equation. The approach effectively transforms an unbounded-domain problem into a bounded-domain formulation, avoiding difficulties associated with unboundedness while maintaining high accuracy. The use of GJP basis functions naturally accommodates the homogeneous boundary conditions and leads to sparse discrete systems that are computationally attractive. The developed framework provides a rigorous theoretical foundation and can be extended to a broader class of nonlinear wave propagation problems.