General splitting methods for abstract semilinear evolution equations

Abstract

In this paper we present a unified picture concerning general splitting methods for solving a large class of semilinear problems: nonlinear Schr¨odinger, Schr¨odinger–Poisson, Gross– Pitaevskii equations, etc. This picture includes as particular instances known schemes such as LieTrotter, Strang, and Ruth–Yoshida. The convergence result is presented in suitable Hilbert spaces related to the time regularity of the solution and is based on Lipschitz estimates for the nonlinearity. In addition, with extra requirements both on the regularity of the initial datum and on the nonlinearity, we show the linear convergence of these methods. We finally mention that in some special cases in which the linear convergence result is known, the assumptions we made are less restrictive.