General splitting methods for abstract semilinear evolution equations

Constanza Sánchez de la Vega (CONICET, Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina; Instituto del Desarrollo Humano, Universidad Nacional de General Sarmiento, Buenos Aires, Argentina)

Abstract

In this paper we present a unified picture concerning general splitting methods for solving a large class of semilinear problems: nonlinear Schrödinger, Schrödinger-Poisson, Gross-Pitaevskii equations, etc. This picture includes as particular instances known schemes such as Lie-Trotter, 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.

Keywords

Lie-Trotter, splitting integrators, semilinear problems

2010 Mathematics Subject Classification

35Q55, 35Q60, 65M12

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Published 16 July 2014