ORDER OF CONVERGENCE OF SPLITTING SCHEMES FOR BOTH DETERMINISTIC AND STOCHASTIC NONLINEAR SCHR?DINGER EQUATIONS?

摘要

We first prove the second order convergence of the Strang-type splitting scheme for the nonlinear Schr?dinger equation. The proof does not require commutator estimates but crucially relies on an integral representation of the scheme. It reveals the connection between Strang-type splitting and the midpoint rule. We then show that the integral representation idea can also be used to study the stochastic nonlinear Schr?dinger equation with multiplicative noise of Stratonovich type. Even though the nonlinear term there is not globally Lipschitz, we prove the first order convergence of a splitting scheme of it. Both schemes preserve the mass. They are very efficient because they use explicit formulas to solve the subproblems containing the nonlinear or the nonlinear plus stochastic terms.