周 麗

(安徽農(nóng)業(yè)大學(xué) 數(shù)學(xué)系， 安徽 合肥 230031)

Kuramato-Tsuzuki方程的有限差分法

周 麗

(安徽農(nóng)業(yè)大學(xué) 數(shù)學(xué)系， 安徽 合肥 230031)

對二維Kuramoto-Tsuzuki方程混合初邊值問題建立了線性化Grank-Nicolson格式，證明了差分格式解存在的唯一性、收斂性，并證明了收斂階為O(τ+h2)。

Kuramoto-Tsuzuki方程; 差分格式; 收斂性

0 引 言

Kuramoto-Tsuzuki方程描述了在歧點(diǎn)附近兩個(gè)分支的行為狀況[1]，文中討論混合初邊值問題的Kuramoto-Tsuzuki方程[2]的數(shù)值解

(1)

(2)

(3)

1 差分格式的建立

對方程(1)-方程(3)建立如下線性化Grank-Nicolson格式：

(4)

(5)

2 差分格式的可解性和收斂性

首先引入下面的Brouwer不動點(diǎn)定理[8-9]:引理1 設(shè)(H,(·,·)H)是有限維內(nèi)積空間，‖·‖H是其上定義的范數(shù)，映射g:H→H是連續(xù)的，若存在α>0,使得對任意z∈H,‖z‖H=α，有Re(g(z),z)H≥0成立，則存在z*∈H,使得‖z*‖≤α?xí)rg(z*)=0。

證明 將方程(4)改寫成:

做映射G:CM+1→CM+1

對上式兩邊同時(shí)取實(shí)部得

解的唯一性用數(shù)學(xué)歸納法可證，證明略。

證明 由于

其中

(6)

(7)

假設(shè)u(x,t)在Ω×(0，T]上有界，則

由引理3知

由引理3

兩邊同時(shí)取實(shí)部得

當(dāng)τ充分小時(shí)，由離散Gronwall不等式得到

命題得證。

[1]YKuramoto，TTsuzuki.Ontheformationofdissipativestructiveinreaction-diffusionsystems[J].ProgrTheorPhys.,1975,54:689-699.

[2] Tingchun Wang, Boling Guo. A robust semi-explicit difference scheme for the Kuramoto-Tsuzuki equation [J]. J.Compu-tational and Applied Mathematics,2009,233:878-888.

[3] Z Z Sun. A linearized difference for the Kuramato-Tsuzuki equation [J]. Comput Math.,1996,14:1-7.

[4] Z Z Sun. OnL∞convergence of a linearized difference scheme for the Kuramato-Tsuzuki equation [J]. Nanjing Univ. J.Math. Biquarterly.，1997,14(1):5-9.

[5] Z Z Sun. A generalized Box scheme for the Kuramato-Tsuzuki equation[J]. J.Southeast Univ.，1996,26(1):87-92.

[6] G Z Tsertsvadze. On tne convergence of a linearized difference scheme for the Kuramato-Tsuzuki equation and for systems of reaction-diffusion type[J]. Zh Vychisl Mat. Fiz.,1991,31:698-707.

[7] Z Z Sun, Q D Zhu. On tsertsvadze’s difference scheme for the Kuramato-Tsuzuki equation[J]. J.Comp., Appl., Math.,1998,98:289-304.

[8] G D Akrivis. Finite difference discretization of the cubic schrodinger equation[J]. IMA J.Numer Anal.,1993,13:115-124.

[9] G D Akrivis, V A Dougalis, O A Karakashian. On fully discrete galerkin methods of second-order temporal accu-racy for the nonlinear schrodinger equation[J]. Numer Math.,1991,59:31-53.

A finite difference scheme for Kuramoto-Tsuzuk equation

ZHOU Li

(Department of Mathematics, Anhui Agricultural University, Hefei 230031， China)

A linearized Crank-Nicolson finite difference scheme is studied for the mixed initial boundary of two-dimensional Kuramoto-Tsuzuki equations. Existence, uniqueness of the solutions are proved and the convergence order isO(τ+h2).

Kuramoto-Tsuzuki equation; difference scheme; convergence.

2014-06-02

安徽農(nóng)業(yè)大學(xué)青年科學(xué)基金資助項(xiàng)目(2011zr007)

周 麗(1981-)，女，漢族，安徽蚌埠人，安徽農(nóng)業(yè)大學(xué)講師，碩士，主要從事偏微分方程的數(shù)值解方向研究,E-mail:lizhou@ahau.edu.cn.

O 241.82

A

1674-1374(2014)05-0585-04