广义非线性Schro¨dinger方程的一个新的守恒差分格式

1、:1000-4424(2006)01-0079-08O1引言非线性S方程在物理学(如非线性光学Q等离子物理学)的应用中扮演()chrodingerNPS着重要的角色,因此最近二十年来对该方程进行了大量的研究R在数值求解方面,也已提出了许多方法,其中包括文S总结和提出了五个有限差分格式,提出了一种三层七点格1TS2T式,提出了US3,4T-V格式和加权格式R本文将讨论更一般的广义非线性Schrodinger方程:()WNPSaY_YYb0,c,0eZgh,dececfXZY(c,Z)bY(c,Z)b0,0eZgh,dfY(c,0)bY(c),c,0dececf这里是实数,是正的实数R,_ai用(

2、式乘以i用(式乘以得到实部部分,则(式1.1)1.1)1.1)j(1.3)Y得到虚部部分,XZ有着如下的关系:(1.1)(1.2)(1.3)收稿日期:2004-02-20基金项目:国家自然科学基金(教育部高等学校博士点基金(10471079);20030422049)dGQ=高校应用数学学报e辑|u|dx=constant,p|u|+|u|E=+2RR第21卷第1期(1.4)dx=constant.A2差分格式及其守恒量对平面区域B做网格剖分,取空间步长时间步长为,EFBG,=(Ix)J,xxHExxKLCDDC为网比记N为原问题其中x=x+M,=N=G,1,O,KPN=1,2,=,uxLLM

3、DMCNM2L(x)精确解u的近似,我们引入一些符号U(1.1)S(1.T)(,)xLMNQN+1N+1N+1NI1NI1NI12NI1uu+uuu+u,Vu=.M+1I2MMI1RM+1I2MMI1RxM2Q2Q(x)(x)对(式提出下面的差分格式U1.1)2N+1Vu=xMN+1NI12N+12NI1u+Vu=pVIxMxMRQLN+1N+1I2NI12N+12NI1I2NI1N+1NI1u|uuu+|uu+u|u.M|+|M|M|+O+|M|M|M|RMMRQQ+2定理X差分格式(关于离散电荷和离散能是守恒的,即2.1)NNI1GQ=Q=O=Q,NNI1GE=E=O=E.其中,NN+1N

4、Q=(YuY2+YuY2)J2,N+1+2N+2E=p(YuY+YuY)J2+x(|u+|u|).x|MZ+2M=GNN+1xNx这里KNN2MNxKYuY=xu|,YuY=xZ|ZM=GN+1NI1和将(式与做内积,然后取2.1)x(u+u)J2分别表示取实部和虚部,虚部部分得到U证其中,N+1NI1N+1NI1N+1NI1_=(uIu(u+u,YuY2IYuY2MM)MM)=Z2ML2L=GKK2N+12NI1N+1NI1=p(Vu+Vu(u+u=G,MM)xMxM)Z2M=GN+1N+1+2N+12NI1+2NI1|uu+O+|uu+|ua=cM|+|M|M|M|M|RQZ2M+2=GK

5、2+2(1.5)R(2.1)(2.2)(2.T)(2.4)K(2.5)M=GNNM+1M.x_+a=G.(2.b)_左进明等:广义非线性方程的一个新的守恒差分格式ZYiabeZn+1n-1n+1n-1(u+u(u+u.jj)jj)=01式中-把,式并令:,的表达式代入(2.6)u表示u的共轭,nn+1nQ=(u2+u2)/2,nn-1则可得Q递推之,即可得到(式.=Q,2.2)n+1n-1将(式与然后取实部部分得:2.1)(/2做内积,xu-u)+=0.其中,n+1n-1n+1n-1=R(u-u(u-u,jj)jj)=02tj=0(2.7)J2n+12n-1n+1n-1=Rp(u+u(u-u=

6、xjxj)jj)2j=0n+1n-1n+1n-1-Rp(u+uu-u=xx,xx)2j=0n+12n-12-p(u/2,x-ux)JJJn+1n+1-2n-12=R(|uuuj|+|j|j|+2+2j=0n+12n-1-2n-1n+1n-1n+1n-1|uu+|u)(u+u(u-uj|j|j|jj)jj)=n+1+2n-1+2x(|u-|u.j|j|)+2j=0把,式中,并令:,代入(2.7)n+1+2n+2E=p(u+u)/2+x(|u+|u|),j|j+2j=0nn-1则可得到E递推之,即可得到(式.=E,2.3)nn+1x2nx2JJO3差分格式的收敛性n当在(式中u的近似是收敛的,为了

7、证明这一点,定P0,P0时,2.1)(,)xtxtj对ujn义如下算子:22jn+1jn-1jn+1jn-1xxQu(x,t)-p+jn2t23S-222(|u(x,t)|+|u(x,t)|u(x,t)|+|u(x,t)|Tjn+1jn+1jn-1jn+1+2-2|u(x,t)|+|u(x,t)|)jn-1jn-1Qu=iuu+|u|u=0.t-p由U展开式得:VWXYZjn+1jn-1,2S(3.1)(3.2)2(u(x,t)(t)+jntt2u(x,t)=u(x,t)+(u(x,t)t+jn+1jnjnt=2高校应用数学学报>辑34(u(x,t)(t)+O(t),jnttt6u(x,

8、t)=u(x,t)-(u(x,t)t+jn-1jnjnt第21卷第1期2,t)(t)-(u(xjntt234(u(x,t)(t)+O(t),jnttt6u(x,t)=u(x,t)+(u(x,t)t+(u(x,t)x+j+1n+1jnjntjnx2(u(x,t)(t)+jntt222(u(x,t)(x)+(u(x,t)xt+(u(x,t)x(t)+jnxxjnxtjnxtt22233(u(x,t)(x)t+(u(x,t)(x)+(u(x,t)(t)+jnxxtjnxxxjnttt26644(O(x)+O(t),u(x,t)=u(x,t)+(u(x,t)t-(u(x,t)x+j-1n+1jnjnt

9、jnx2(u(x,t)(t)+jntt222(u(x,t)(x)-(u(x,t)xt-(u(x,t)x(t)+jnxxjnxtjnxtt22233,t)(x)t-(,t)(x)+(,t)(t)+(u(xu(xu(xjnxxtjnxxxjnttt26644(O(x)+O(t),t)=u(x,t)-(u(x,t)t+(u(x,t)x+u(xj+1n-1jnjntjnx2(u(x,t)(t)+jntt222(u(x,t)(x)-(u(x,t)xt+(u(x,t)x(t)-jnxxjnxtjnxtt22233(u(x,t)(x)t+(u(x,t)(x)-(u(x,t)(t)+jnxxtjnxxxjnt

10、tt26644(O(x)+O(t),u(x,t)=u(x,t)-(u(x,t)t-(u(x,t)x+j-1n-1jnjntjnx2(u(x,t)(t)+jntt222(u(x,t)(x)+(u(x,t)xt-(u(x,t)x(t)-jnxxjnxtjnxtt22233(u(x,t)(x)t-(u(x,t)(x)-(u(x,t)(t)+jnxxtjnxxxjnttt26644(O(x)+O(t).则差分格式的截断误差为:322244Lu(x,t)-Lu=u(t)-u(t)-u(x)+(O(x)+O(t).jntttxxttxxxx62123故在网格比确定的情况下,当该问题是收敛的.0,0时,L(

11、,)-L0,xtuxtujn<4差分格式的稳定性构造格式时需要证明计算过程中的误差影响,如果计算过程的误差保持有界或消失,就U左进明等:广义非线性R方程的一个新的守恒差分格式cSToViWXeTQ3说明它是稳定的,否则就是不稳定的.如果格式不稳定,即使收敛性是满足的,这个方案也是无用的.2让r则(可转化为:=t/(x),2.1)n+1n-1n+1n+1n+1n-1n-1n-1i(u-u=pr(uu+uu+u)-jj)j+1-2jj-1+uj+1-2jj-1n+1-2n-12n+12n-1-2n-1n+1n-1|uuuu+|u)(u+u,j|j|+|j|j|j|jj)即n+1(t)(|uj

12、|+2(4.1)i+2pr+n+1n+1-2n-12n+12n-1-2(t)(|uuuuu+j|+|j|j|+|j|j|+2n-1n+1)u+-i+2pr+|uj|jn+1n+1-2n-12(t)(|uuuj|+|j|j|+2n+12n-1-2n-1n-1n+1n+1n-1n-1|uu+|u)u-pr(u)=0.j|j|j|jj+1+uj-1+uj+1+uj-1nnixixj代换u并除去e得到=Vej,j(4.2)i+2pr+n+1n+1n-12n+12n-1-2(t)(|V|+|V|-2|V|+|V|V|+2n-1n+1|V|V+-i+2pr+n+1n+1-2n-12(t)(|V|+|V|V

13、|+2n+12n-1n-1n-1n+1n-1|V|-2+|V|V-2prcos(x)(V+V)=0.|V(4.3)n-1n+1对于项V从(式可以看出它们的系数是递减的,由稳定性的定义,4.3)V是有界的,可知该差分格式是稳定的.HI数值分析式两边同乘以则可转化为:(2.1),tn+1n-1n+1n+1n+1n-1n-1n-1i(u-u-pr(uu+uu+u)+jj)j+1-2jj-1+uj+1-2jj-1n+1(t)(|uj|+2n+1-2n-12n+12n-1-2n-1n+1n-1|uuuu+|u)(u+u=0,j|j|+|j|j|j|jj)那么可得到下面的线性方程组:n+1n+1n+1n+

14、1n+1n+1n+1+LJjuj+1+Kjujjuj-1=Mj.n+1其中,J=-pr,jn+1K=i+2pr+jn+1L=-pr,jn+1n-1M=prui-2pr-jj+1+(I.1)(I.2)n+1n+1-2n-12n+12n-1-2n-1(t)(|uuuuu+|u),j|+|j|j|+|j|j|j|+2n+1n+1-2n-12(t)(|uuuj|+|j|j|+2n+12n-1-2n-1n-1n-1|uu+|u)u+pru.j|j|j|jj-1将(式写成矩阵的形式I.2)NO=P,(I.3)kZ其中,高校应用数学学报l辑第21卷第1期1BA1C=T=BN-1A0.B2=A=CBNN

15、15;NTn+1n+1n+1n+1,u,dQ=uD=d1,N,1,N.11n+1n+1-2n-12到u然后用本文的差分格式来处理即可.对于项|,uuujj|+|j|j|+n+12n-1-2n-1n-1nn+1n+1nn-1我们可用u的外插对u|uu+|uu=2u,j|j|j|,j和ujj进行赋值:jj+uj就可以得到理想的结果.LM数值算例现在我们将考虑如下算例:取u=-1,=2,=2.(,0)=Q(+10)2(+NOPRSTPRUVWP则该问题的孤立子解为:10),=-P=15,PXYu(P,t)=QRST(P+10-Zt)RUV2W(P+10)-3Wt.(M.1)取取式精确解差的范数,列于

16、=0.1,=0.05,0.01两种情况来计算数值解和(M.1)表1和表2其中包括文中提到的格式,文中提出的一种守恒(1-()2_abWScdQcb)的e格式和本文提出的守恒e对称格式)-:TRRdRfRdgRfRVcWhTRRdRfRdgWUVcWh表1=0.1,=0.05nt-_abWScdQc-eTRRdRfRdgRfRVcWh2-eTRRdRfRdgWUVcWh左进明等:广义非线性S方程的一个新的守恒差分格式chrodinger表2h=0.1,=0.01nt85-CrankNicolson1-ThreelevelSevenpoint2-ThreelevelSixpoint从表中可以看出,

17、本文的格式优于上面的C-rankNicolson格式和ThreelevelSeven格式.而且本文的格式是守恒的,对于非守恒的格式,在计算中容易出现非线性的"爆point2,5现象炸".参考文献:1T,Ab.AnahaTRlowitzMJalyticalandnumericalaspectsofcertainnonlinearevolution,.,1984,55:203-230.equationsIINumericalNonlinearSchrodingerEquationJJComputPhys2Z,P-.NuhangFeiérezGgarciaVM,Vdzq

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