盛秀兰,冯美娇,吴宏伟
(1.东南大学数学系,南京210096;2.江苏开放大学,南京210036)
变系数zakharov-Kuznetsov方程的三层线性隐式差分格式
盛秀兰1,2*,冯美娇1,吴宏伟1
(1.东南大学数学系,南京210096;2.江苏开放大学,南京210036)
利用有限差分法逼近变系数广义ZK(Zakharov-Kuznetsov)方程的初边值问题,建构一个三层线性化隐式差分格式.利用离散能量估计方法,讨论差分格式解的唯一性以及x方向的一阶差商在L∞模意义下的收敛性、稳定性和收敛阶数,并通过数值算例验证理论分析的结果.
Zakharov-Kuznetsov方程;隐式差分格式;收敛性;稳定性
本文研究下列Zakharov-Kuznetsov方程初边值问题的数值解法:
其中Ω=(L,R)×(L,R);g(u,t)=α(t)u+β(t)u2,式中α(t),β(t)和γ(t)是关于t的任意函数;φ1(x,t),φ2(x,t),φ(x,y)是已知光滑函数,且满足相容性条件.Abdou等[1]用映射的方法给出了ZK方程的周期解.Ganjavi等[2]用同伦摄动法及变分迭代法给出了ZK方程的解.Yan等[3]结合李对称群给出了ZK方程的对称相似解.Wang[4],Bustamante[5],Ribaud[6]等从理论上研究了ZK方程解的存在唯一性.Ma等[7]给出了一种借助辅助方程求ZK方程精确解的方法.然而,利用有限差分法求ZK方程问题数值解的研究很少.考虑到ZK方程的特点,可从色散方程方面探讨其数值解法.Darvishi[8]和 Haq[9]等分别利用谱配置方法、Darvishi预条件法和无网格方法研究了 Kdv-Burgers方程的数值解法.Nishiyama等[10]研究了二维广义ZK方程的守恒有限差分格式及数值稳定性.本文主要利用有限差分法研究问题(1)~(3)式,对非线性项g(u,t)ux文献[11]给出了一种两层线性化的差分格式,其他相关方法可参考文献[12].
设存在常数C0,且有如下假设:
C0+S,即当l=k+1时,由归纳法证得定理2成立.
差分格式(4)~(6)式在实际运算过程中以时间逐层计算,误差是不可避免的,从而直接影响差分格式的稳定性.类似差分格式收敛性的验证,可得到其稳定性.
利用差分格式计算了一个实例,计算结果见表1.表中列出了不同步长时数值解的最大误差和误差比,其中
其中f(x,y,t)=ey+tsin(2πx){1+[1+ ey+tsin(2πx)]·2πey+tcos(2πx)},该问题的精确解为u(x,y,t)=ey+tsin(2πx).
从表1可以看出,当步长缩小为原来的1/2时,误差约缩小为原来的1/4,同时数值解也很好地逼近精确解.这与本文分析的结果吻合,即差分格式的解在L∞模下的收敛阶数为O(h2+τ2).
表1 差分解在不同步长下的最大误差和误差比Tab.1 Errors and convergence rate
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A three-level difference scheme for zakharov-Kuznetsov equation
SHENG Xiulan1,2*,FENG Meijiao1,WU Hongwei1
(1.Dept of Math,Southeast Univ,Nanjing 210096,China;2.Jiangsu Open Univ,Nanjing 210036,China)
In this paper,by using finite difference method,an implicit difference scheme is constructed to approximate the initial-boundary value problem of ZK equation.The proposed scheme is a three-level linearization scheme.Using the method of discrete energy estimates,existence uniqueness of difference scheme is proved.With the method of the discrete energy estimate,it is shown that the difference scheme is convergent in maximum norm.The convergence order is second-order in both space and time.Some numerical experiments are conducted to illustrate the theoretical results of the proposed difference scheme.
Zakharov-Kuznetsov equation;implicit difference scheme;convergence;stability
O241.82
A
1007-824X(2015)02-0031-04
(责任编辑 何青玉)
2014-07-14.*联系人,E-mail:113525336@qq.com.
国家自然科学基金资助项目(11271068);江苏开放大学“十二五”规划课题(13SEW-C-076).
盛秀兰,冯美娇,吴宏伟.变系数Zakharov-Kuznetsov方程的三层线性隐式差分格式[J].扬州大学学报:自然科学版,2015,18(2):31-34,39.