摘要
在量子力学与高能物理中,非线性Schrodinger方程很重要,它和KdV方程、BBM方程及Sine-Gordon方程一样,早就引起了人们的注意.郭柏灵讨论了非线性Sch-rodinger方程的适定性与数值方法;吴相辉研究了四点和六点隐差分格式的收敛性和稳定性;常谦顺探讨了守恒差分格式.
In this paper, the following class of non-self-adjoint and nonlinear Schrodin getequation is considered: iU_t-(α(x,t)U_x)_x+b(x,t)?_x+c(x,t)q(|U|~2)U+d(x,t)U=f(x,t), -∞<x<+∞,0<t<T; U(x,0)=U_0(x),-∞<x<+∞; U(x+2π,t)=U(x,t),x∈R,t≥0.A leap-frog finite difference scheme is given,and its convergence and stability areproved.The following result is obtained: Theorem.Assume that U∈C^4,and ‖U^0-u^0‖+‖U^1-u^1‖+max 0≤r≤[T/k]‖?‖=o(h^1/2).If there exists a constant σ with 4MS +KC≤σ<1,then there exist positive con-stants C_m(m=1,2,3)independent of k and h such that,for k≤C_1,h≤C_2, ‖U^n-u^n‖≤C_3(k^2+h^2+‖U^0-u^0+‖U^1-u^1‖+max 0≤r≤[T/k]‖?‖)where S=kh^(-2),C=kh^(-1),M=max 0≤x≤2π 0≤t≤T |α(x,t)|,K=max 0≤x≤2π 0≤t≤T|b(x,t)|. Finally,it is shown by a numerical example that numerical results are coinci-dent with theoretical results.
出处
《计算数学》
CSCD
北大核心
1989年第2期118-127,共10页
Mathematica Numerica Sinica
