General splitting methods for abstract semilinear evolution equations

J. P. BORGNA; M. DE LEO; CONSTANZA S. F. DE LA VEGA; D. RIAL

COMMUNICATIONS IN MATHEMATICAL SCIENCES

INT PRESS BOSTON, INC

Año: 2015 vol. 13 p. 83 - 101

1539-6746

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 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 nonlin- earity, 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.