具波动算子非线性Schr?dinger方程的一种守恒差分格式
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摘要
研究了一类具波动算子的非线性Schr?dinger方程的数值计算问题.给出了该方程的两个守恒律,构造了求解该方程近似解的一种守恒差分格式,使该差分格式的精度在时间和空间上均达到二阶精度,并对该格式的收敛性及稳定性进行了证明.数值实验与理论结果相一致,很好地验证了本文提出的离散格式.
The numerical calculation of nonlinear Schr?dinger equation with wave operator was studied,which contains two conservation laws,and a conservative difference scheme with two orders accuracy in both time and space.Meanwhile,the convergence and stability were proved.Finally,the discrete format proposed in this paper was examined by the numerical simulation,the numerical experiments gave a good agreement with the theoretical results.
The numerical calculation of nonlinear Schr?dinger equation with wave operator was studied,which contains two conservation laws,and a conservative difference scheme with two orders accuracy in both time and space.Meanwhile,the convergence and stability were proved.Finally,the discrete format proposed in this paper was examined by the numerical simulation,the numerical experiments gave a good agreement with the theoretical results.
引文
[1]MATSUUCHI K.Nonlinear interactions of counter-travelling waves[J].J Phy Soc Jap,1980,48(5):1746-1754.
[2]BERGE L,COLIN T.A singular perturbation problem for an envelope equation in plasma physics[J].Physica D:Nonlinear Phenomena,1995,84(3):437-459.
[3]BAO W Z,DONG X C,JACK X.Comparisons between Sine-Gordon and perturbed nonlinear Schr9dinger equations for modeling light bullets beyond critical collapse[J].Physica D:Nonlinear Phenomena,2010,239(13):1120-1134.
[4]MACHIHARA S,NAKANISHI K,OZAWA T.Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations[J].Math Ann,2002,322(3):603-621
[5]郭柏灵,梁华湘.具波动算子的一类非线性Schr9dinger方程组的数值计算问题[J].数值计算与计算机应用,1983,4(3):176-182.
[6]陈娟.一类非线性Schr9dinger方程的Jacobi椭圆函数周期解[J].应用数学学报,2014,37(4):656-661.
[7]ZHANG L M,CHANG Q S.A conservative numerical scheme for a class of nonlinear Schr9dinger equation with wave operator[J].Appl Math Comput,2003,145(2):603-612.
[8]WANG T C,ZHANG L M.Analysis of some new conservative schemes for nonlinear Schr9dinger equation with wave operator[J].Appl Math Comput,2006,182(182):1780-1794.
[9]胡汉章,谢水连.一类带波动算子的非线性Schr9dinger方程的高精度守恒差分格式[J].高校应用数学学报,2014,29(1):36-44.
[10]ZHANG L M,LI X G.A conservative finite difference scheme for a class of nonlinear Schr9dinger equation with wave operator[J].Acta Math Sci,2001,22A(2):258-263.
[11]LI G,YAN X.Energy conserving local discontinuous galerkin methods for the nonlinear Schr9dinger equation with wave operator[J].J Sci Comput,2015,65(2):622-647.
[12]郭柏灵.具波动算子的一类多维非线性Schr9dinger方程组的初、边值问题[J].中国科学(A辑),1983,26(2):134-146
[13]梁宗旗,鲁百年.具有波动算子的非线性Schr9dinger方程的拟谱方法[J].高等学校计算数学学报,1999,9(3):201-211.
[14]梁宗旗,张发勇.具有波动算子的非线性Schr9dinger方程的有限差分方法[J].黑龙江大学学报(自然科学版),1983,15(1):1-4.
[15]梁宗旗.具有波动算子的非线性Schr9dinger方程的谱方法[J].陕西师范大学学报(自然科学版),2002,16(2):9-15.
[16]WANG J.Multisymplectic fourier pseudospectral method for the nonlinear Schr9dinger equation with wave operator[J].JComput Math,2007,25(1):31-48.
[17]KONG L H,WAN L,YIN X L,et al.Conservative properties analysis of multi-symplectic integrator for the Schr9dinger equation with wave operator[J/OL].Numerical Analysis and Applied Mathematics,2012[2017-12-02].DOI:10.1063/1.4756389.
[18]ZHANG F,PEREZ-ARCIA M V,VAZQUEZ L.Numerical simulation of nonlinear Schr9dinger equation system:a new conservative scheme[J].Appli Math Comput,1995,71(2):165-177.
[19]CHAN T F,SHEN L J.Stability analysis of difference schemes for variable coefficient Schrodinger type equations[J].SIAM JNumer Anal,1981(24):336-349.
[20]GUO B Y,PASCUAL P J,RODRIGUEZ M J,et al.Numerical solution of the Sine-Gordon equation[J].Appl Math Comput,1986,18(1):1-14.
[2]BERGE L,COLIN T.A singular perturbation problem for an envelope equation in plasma physics[J].Physica D:Nonlinear Phenomena,1995,84(3):437-459.
[3]BAO W Z,DONG X C,JACK X.Comparisons between Sine-Gordon and perturbed nonlinear Schr9dinger equations for modeling light bullets beyond critical collapse[J].Physica D:Nonlinear Phenomena,2010,239(13):1120-1134.
[4]MACHIHARA S,NAKANISHI K,OZAWA T.Nonrelativistic limit in the energy space for nonlinear Klein-Gordon equations[J].Math Ann,2002,322(3):603-621
[5]郭柏灵,梁华湘.具波动算子的一类非线性Schr9dinger方程组的数值计算问题[J].数值计算与计算机应用,1983,4(3):176-182.
[6]陈娟.一类非线性Schr9dinger方程的Jacobi椭圆函数周期解[J].应用数学学报,2014,37(4):656-661.
[7]ZHANG L M,CHANG Q S.A conservative numerical scheme for a class of nonlinear Schr9dinger equation with wave operator[J].Appl Math Comput,2003,145(2):603-612.
[8]WANG T C,ZHANG L M.Analysis of some new conservative schemes for nonlinear Schr9dinger equation with wave operator[J].Appl Math Comput,2006,182(182):1780-1794.
[9]胡汉章,谢水连.一类带波动算子的非线性Schr9dinger方程的高精度守恒差分格式[J].高校应用数学学报,2014,29(1):36-44.
[10]ZHANG L M,LI X G.A conservative finite difference scheme for a class of nonlinear Schr9dinger equation with wave operator[J].Acta Math Sci,2001,22A(2):258-263.
[11]LI G,YAN X.Energy conserving local discontinuous galerkin methods for the nonlinear Schr9dinger equation with wave operator[J].J Sci Comput,2015,65(2):622-647.
[12]郭柏灵.具波动算子的一类多维非线性Schr9dinger方程组的初、边值问题[J].中国科学(A辑),1983,26(2):134-146
[13]梁宗旗,鲁百年.具有波动算子的非线性Schr9dinger方程的拟谱方法[J].高等学校计算数学学报,1999,9(3):201-211.
[14]梁宗旗,张发勇.具有波动算子的非线性Schr9dinger方程的有限差分方法[J].黑龙江大学学报(自然科学版),1983,15(1):1-4.
[15]梁宗旗.具有波动算子的非线性Schr9dinger方程的谱方法[J].陕西师范大学学报(自然科学版),2002,16(2):9-15.
[16]WANG J.Multisymplectic fourier pseudospectral method for the nonlinear Schr9dinger equation with wave operator[J].JComput Math,2007,25(1):31-48.
[17]KONG L H,WAN L,YIN X L,et al.Conservative properties analysis of multi-symplectic integrator for the Schr9dinger equation with wave operator[J/OL].Numerical Analysis and Applied Mathematics,2012[2017-12-02].DOI:10.1063/1.4756389.
[18]ZHANG F,PEREZ-ARCIA M V,VAZQUEZ L.Numerical simulation of nonlinear Schr9dinger equation system:a new conservative scheme[J].Appli Math Comput,1995,71(2):165-177.
[19]CHAN T F,SHEN L J.Stability analysis of difference schemes for variable coefficient Schrodinger type equations[J].SIAM JNumer Anal,1981(24):336-349.
[20]GUO B Y,PASCUAL P J,RODRIGUEZ M J,et al.Numerical solution of the Sine-Gordon equation[J].Appl Math Comput,1986,18(1):1-14.