Order estimates in time of splitting methods for the nonlinear Schrödinger equation. (English) Zbl 1026.65073

Besse, Christophe; Bidégaray, Brigitte; Descombes, Stéphane

SIAM J. Numer. Anal. 40, No. 1, 26-40 (2002).

The nonlinear Schrödinger equation \[ u_t+ i\Delta u- F(u)= 0, \] for \(t> 0\) in two dimension and under the initial conditions \(u(x,0)= u_0(x)\) is investigated. By an operator-theoretic proof it is shown, that the Lie and Strang formulas for splitting [cf. G. Strang, ibid. 5, 506-517 (1968; Zbl 0184.38503)] are the approximations of the exact solution of the given equation, which are of the order 1 and 2 in time.

MSC:

65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35Q55	NLS equations (nonlinear Schrödinger equations)
65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs

Keywords:

nonlinear Schrödinger equation; splitting methods; Lieformula; difference schemes