A new linearized second-order energy-stable finite element scheme for the nonlinear Benjamin-Bona-Mahony-Burgers equation

In this paper, a novel linearized second-order energy-stable fully discrete scheme for the nonlinear Benjamin-Bona-Mahony-Burgers (BBMB) equation is introduced, along with a superconvergence analysis of the conforming finite element method (FEM). Through skillful decomposition of the nonlinear term u∇⋅u, a new linearized second-order fully discrete scheme is developed. This scheme requires fewer iterations and exhibits higher computational efficiency compared to the nonlinear approach. Furthermore, it maintains energy stability, enabling direct numerical solution boundedness in the H1-norm, which represents an improvement over the L∞-norm boundedness reported in previous literature. Secondly, relying on this boundedness and the high-precision results of the bilinear element, we derive the unconditional error estimates for superclose and global superconvergence without any restrictions on the time step size Δt and spatial mesh size h. Finally, numerical examples are presented to validate the theoretical analysis.