Order of Convergence of Splitting Schemes for Both Deterministic and Stochastic Nonlinear Schrödinger Equations

Abstract

We first prove the second order convergence of the Strang-type splitting scheme for the nonlinear Schrödinger 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 Schrödinger 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.

Analysis of a Splitting Scheme for Damped Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise

“…Iterating previous procedures, we obtain a splitting process u τ = {u τ (t) : t ∈ [0, T ]}, which is left-continuous with finite right-hand limits and F t -adapted. We note that there are some results on numerically approximating SPDEs by splitting schemes (see [10,13,19,21,26] and references therein). Since (5) has no analytic solution, we apply the Crank-Nicolson type scheme to temporally discretize (5).…”

Convergence of a $$\theta $$ θ -scheme to solve the stochastic nonlinear Schrödinger equation with Stratonovich noise

“…This property is not known for the Crank-Nicolson scheme (θ = 1 2 ), which is why a truncation strategy is applied to the nonlinearity (see [7]) or the noise term ( [3]), leading to a truncated Crank-Nicolson scheme. The next lemma establishes this property for the θ -scheme and values θ ∈…”

“…A further step towards constructing efficient discretizations of (1) is the work [7] which uses a Lie-type time-splitting method. This scheme amounts to solving a family of timely explicitly discretized SODEs for all x ∈ R d , and a linear PDE with random force.…”

New scheme for nonlinear Schrödinger equations with variable coefficients

“…A semi-discrete scheme is considered to NSE with power nonlinearity [5]. The Strang-type splitting scheme is proposed to NSE with multiplicative noise [13]. A θ-scheme is analyzed for NSE with Stratonovich noise [3].…”