Combined Method for Solving of 1D Nonlinear Schrödinger Equation
- 943 Downloads
We propose combined method, based on using of both the conservative finite-difference scheme and non-conservative Rosenbrock method, for solving a linear or non-linear 1D Schrödinger equation. The computer simulation results, obtained by using of combined method, are compared with corresponding results obtained using the conservative finite-difference scheme or Rosenbrock method. For 2D nonlinear problem the proposed method can significantly increase a computer simulation performance due to eliminating of using an iterative process, which is necessary for the conservative finite-difference scheme realization. The efficiency of this combined method with artificial boundary conditions is demonstrated by numerical experiments.
KeywordsRosenbrock method Conservative finite-difference scheme Schrödinger equation Artificial boundary conditions
This work was supported in part by the Russian Foundation for Basic Research under Grant 12-01-00682-a.
- 2.Karamzin YuN, Sukhorukov AP, Trofimov VA (1989) Mathematical modeling in nonlinear optics. Moskva: Izd-vo Moskovskogo universitetaGoogle Scholar
- 6.Dnestrovskaya EYu, Kalitkin NN, Ritus IV (1991) The solution of partial differential equations by schemes with complex coefficients. Journal of Mathematical Models and Computer Simulations 3:9, pp 114–127Google Scholar
- 8.Gerisch A, Chaplain MAJ (2006) Robust numerical methods for taxis–diffusion–reaction systems: Applications to biomedical problems. Mathematical and Computer Modelling 43:49–75Google Scholar
- 9.Verwer JG, Hundsdorfer WH, Blom JG (1998) Numerical time integration for air pollution models. Modelling, Analysis and Simulation MAS-R9825, pp 1–60Google Scholar
- 20.Zheng C (2006) Exact nonreflecting boundary conditions for one–dimensional cubic nonlinear Schrödinger equations. J Comput Phys 215, pp:552–565Google Scholar
- 21.Xuand Z, Han H (2006) Absorbing boundary conditions for nonlinear Schrödinger equations. Phys Rev 74:037704Google Scholar
- 22.Tereshin EB, Trofimov VA, Fedotov MV (2006) Conservative finite difference scheme for the problem of propagation of a femtosecond pulse in a nonlinear photonic crystal with non-reflecting boundary conditions. Computational Mathematics and Mathematical Physics 46:1, pp 154–164Google Scholar
- 23.Trofimov VA, Dogadushkin PV (2007) Boundary conditions for the problem of femtosecond pulse propagation in absorption layered structure. Farago I, Vabishevich P, Vulkov L, Proceedings of Fourth Intern Conf Finite Difference Methods: Theory and applications, Rouse University Angel Kanchev, Lozenetz, Bulgaria, pp 307–313Google Scholar