Finite difference discretization of the cubic Schrödinger equation.

*(English)*Zbl 0762.65070The initial-boundary value problem for the cubic Schrödinger equation in one space dimension is investigated in the framework in the finite difference approach. The author proposes a Crank-Nicolson type discretization for which he proves the existence theorem and establishes error estimates as well as some conservation and convergence properties.

The essential tools used are a Brouwer-type fixed point theorem and the discrete version of Gronwall’s inequality. The corresponding nonlinear discrete equations are then linearized at each time level by Newton’s method and solved by an algorithm developed for this purpose, for which an error estimate is established.

Reviewer: O.Titow (Berlin)

##### MSC:

65Z05 | Applications of numerical analysis to physics |

65M06 | Finite difference methods (IVP of PDE) |

65M12 | Stability and convergence of numerical methods (IVP of PDE) |

65M15 | Error bounds (IVP of PDE) |

35Q55 | NLS-like (nonlinear Schrödinger) equations |