Convergence and superconvergence analysis of energy-preserving Crank–Nicolson Galerkin method for the Benjamin–Bona–Mahony equation

Abstract

In this paper, the convergence and superconvergence error estimates are investigated for Benjamin–Bona–Mahony equation with energy-preserving Crank–Nicolson Galerkin scheme. Firstly, a priori bound of the numerical solution is obtained due to the energy conservation property of the numerical scheme. Secondly, the existence and uniqueness of the numerical solution is proved rigorously. Then, the superclose error estimate between the numerical solution and Ritz projection of the exact solution in $L^{\mathrm{\infty }}(H^1)$-norm is derived by skilfully dealing with the nonlinear term. Moreover, the superclose error estimate between numerical solution and interpolation of the exact solution in $L^{\mathrm{\infty }}(H^1)$-norm is acquired with a novel approach, which is the combination of the interpolation operator and Ritz projection operator as well as high accuracy error estimate of the bilinear element. Furthermore, the superconvergence error estimate in $L^{\mathrm{\infty }}(H^1)$-norm is obtained with a simple and efficient interpolation post-processing approach. Finally, some numerical results are provided to verify the theoretical analysis.

Keywords:

• Benjamin–Bona–Mahony equation
• energy-preserving property
• Crank–Nicolson Galerkin scheme
• bilinear element
• superclose and superconvergence error estimates

2010 Maths Classifications:

• 65N15
• 65N30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [grant number 12101568].