Analysis of A Splitting Scheme for Damped Stochastic Nonlinear Schrödinger Equation with Multiplicative Noise
This provides rigorous convergence guarantees for numerical simulation of stochastic PDEs with damping, which is relevant for applications in physics and engineering.
The authors prove that for the damped stochastic nonlinear Schrödinger equation with multiplicative noise, a splitting scheme achieves strong order 1/2 and weak order 1, both independent of time, under a large damping condition.
In this paper, we investigate the damped stochastic nonlinear Schrödinger(NLS) equation with multiplicative noise and its splitting-based approximation. When the damped effect is large enough, we prove that the solutions of the damped stochastic NLS equation and the splitting scheme are exponential stable and possess some exponential integrability. These properties lead that the strong order of the scheme is $\frac 12$ and independent of time. Meanwhile, we analyze the regularity of the Kolmogorov equation with respect to the equation. As a consequence, the weak order of the scheme is shown to be twice the strong order and independent of time.