A Two-level Linearized Conserved Difference Algorithm For Solving BBM Equations

In this paper,the finite difference method is used to study the initial and boundary value problems of a class of BBM equations with nonlinear diffusion term and dissipative term.Firstly,the nonlinear terms uux are discretized by two layers of linearization in the time layer,and a two-layer linear difference scheme with O(?2+h2)theoretical accuracy is proposed.Then,Richardson's idea of extrapolation is used to extrapolate the spatial layer to obtain the fourth order theoretical accuracy.Furthermore,a two-level linear difference scheme with O(?2+h4)theoretical accuracy is constructed,and both schemes reasonably simulate a conservation property of the original problem.The convergence and stability of these two difference schemes are proved directly by means of mathematical induction and discrete functional analysis when the maximum modulus estimation of the difference decomposition cannot be obtained.Under the premise of higher theoretical accuracy,the linear difference scheme reduces the storage of computer data in the time layer during numerical solution,thus improving the calculation efficiency.Numerical results show that the scheme is reliable,and its accuracy is obviously better than the general spatial fourth-order difference scheme.