求解BBM方程的一个新的高精度线性化差分算法

A new high accuracy linearized difference algorithm for the BBM equation

摘要:

本文对一类带有齐次边界条件的Benjamin-Bona-Mahony方程的初边值问题进行了数值研究.通过先在时间层外推对问题进行线性化离散处理，然后再利用Richardson外推的思想在空间层进行外推，从本文提出了一个三层线性差分格式，证明了差分解的存在唯一性.在无法得问题差分解的最大模估计的情况下，本文综合运用数学归纳法和离散泛函分析方法直接证明了该差分格式的收敛性和稳定性.数值实验表明该格式的精度明显优于已有的线性层差分格式.

Abstract:

In this paper, an initial boundary value problem of Benjamin-Bona-Mahony equation is studied numerically. Firstly, the problem is discretized and linearized by extrapolating in time layer, and then extrapolated at the space level with Richardson's extrapolation idea, thus a three-layer linear difference scheme is proposed. Secondly, the existence and uniqueness of the decomposition of difference of the discrete solution are proved. Finally, the convergence and stability of the scheme are proved by combining mathematical induction and discrete functional analysis. Numerical experiments show that the accuracy of the scheme is obviously better than that of the known linear layer difference scheme.