Lie-Trotter method for abstract semilinear evolution equations
For researchers working on numerical methods for semilinear PDEs, this work offers a theoretical foundation for splitting methods, but it is largely an incremental extension of existing convergence theories.
The paper provides a unified convergence analysis of the Lie-Trotter splitting method for a broad class of semilinear evolution equations, including nonlinear Schrödinger and Gross-Pitaevskii equations, establishing linear convergence under additional regularity conditions.
In this paper we present a unified picture concerning Lie-Trotter method for solving a large class of semilinear problems: nonlinear Schrödinger, Schröginger--Poisson, Gross--Pitaevskii, etc. This picture includes more general schemes such as Strang and Ruth--Yoshida. The convergence result is presented in suitable Hilbert spaces related with 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 the method.