Linearized difference schemes for a BBM equation with a fractional nonlocal viscous term

Abstract

This paper is concerned with the efficient finite difference schemes for a Benjamin–Bona–Mahony equation with a fractional nonlocal viscous term. By using the weighted-shift Grünwald–Letnikov and the fractional centered difference formulae to approximate the nonlocal fractional operators, we design a class of linearized finite difference schemes for the presented nonlocal model. The existence, stability and convergence of the proposed numerical schemes are rigorously derived with the help of functional analysis. Theoretical analysis shows that the proposed numerical schemes are stable with second order accuracy. Numerical examples are presented to verify our theoretical analysis and to demonstrate the efficiency of the proposed numerical schemes.