[1]张 俊,范馨月.一类浅水波模型的数值方法[J].南京师大学报(自然科学版),2015,38(01):32.
　Zhang Jun,Fan Xinyue.Numerical Methods for a Class of Shallow Water Equation[J].Journal of Nanjing Normal University(Natural Science Edition),2015,38(01):32.
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一类浅水波模型的数值方法()
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《南京师大学报》（自然科学版）[ISSN:1001-4616/CN:32-1239/N]

卷:
第38卷
期数:
2015年01期
页码:
32
栏目:
数学
出版日期:
2015-06-30

文章信息/Info

Title:
Numerical Methods for a Class of Shallow Water Equation
作者:
张 俊1; 范馨月2
(1.贵州财经大学数学与统计学院,贵州 贵阳 550025)(2.贵州大学理学院,贵州 贵阳 550025)
Author(s):
Zhang Jun1; Fan Xinyue2
(1.School of Mathematics and Statistics,Guizhou University of Finance and Economics,Guiyang 550025,China)(2.College of Science,Guizhou University,Guiyang 550025,China)
关键词:
BBM方程; 无条件稳定; 有限差分法; 谱方法; 衰减率
Keywords:
BBM equation; unconditionally stable; finite difference method; spectral method; decay rate
分类号:
O156.5
文献标志码:
A
摘要:
考虑BBM型非线性水波方程的数值方法. 本文构造了二种半隐的数值格式. 以BBM方程为例,严格分析了二种格式的稳定性与误差估计,证明了二种格式都是无条件稳定的. 误差估计显示,线性Euler时间离散加谱Galerkin空间离散的收敛阶是O(Δt+N1-m),线性Crank-Nicolson时间离散加谱Galerkin空间离散的收敛阶是O(Δt2+N1-m). 最后我们用数值例子讨论这两类方程解的长时间衰减率,并讨论扩散项、色散项、非线性项对解的衰减率的影响. 数值例子表明,这两类浅水波方程的衰减率是:L2范接近-1/4; L∞范接近-1/2; H1半范接近-3/4,这与已知的理论结果是吻合的.
Abstract:
We turn to study the numerical solution of the shallow water equation. We propose two different schemes to numerically solve this equation. A detailed analysis is carried out for these schemes,and we prove that the overall schemes are unconditionally stable. The error estimation shows that the linearized Euler schema in time plus Fourier spectral method in space is convergent with the convergence order O(Δt+N1-m),and higher order convergences can be obtained if the second order backward differentiation or Crank-Nicolson schema are used to discretize the equation in time. At last,we use the proposed methods to investigate the asymptotical decay rate of the solutions to the shallow water wave equation. We equally discuss the role of the diffusion terms,the geometric dispersion and the nonlinearity respectively. The performed numerical experiment confirms that the decay rates in L2-norm,L∞-norm,and H1-seminorm are very close to -1/4,-1/2,and -3/4 respectively. These numerical results are consistent with the known theoretical prediction.

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备注/Memo

备注/Memo:
收稿日期:2014-05-16.
基金项目:2013贵州财经大学引进人才项目、贵州省科学技术基金(黔科合J字[2013]2028号).
通讯联系人:张俊,博士,副教授,研究方向:计算数学. E-mail zj654440@163.com
更新日期/Last Update: 2015-03-30