Optimal convergence analysis of an energy dissipation property virtual element method for the nonlinear Benjamin-Bona-Mahony-Burgers equation

A novel arbitrary high-order energy-stable fully discrete schemes are proposed for the nonlinear Benjamin-Bona-Mahony-Burgers equation based on linearized Crank-Nicolson scheme in time and the virtual element discretization in space. Two skew-symmetric discrete forms are introduced to preserve energy dissipation of the numerical scheme. Furthermore, by utilizing the $L^2$ projection to approximate the nonlinear term and estimating the error of the discrete bilinear forms carefully, the optimal error estimate of the numerical scheme in the $H^1$-norm is obtained. Finally, several numerical examples on various mesh types are provided to demonstrate the energy stability, optimal convergence and high efficiency of the method.