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Abstract In this paper, a higher order time-discretization scheme is proposed, where the iterates approximate the solution of the stochastic semilinear wave equation driven by multiplicative noise with general drift and diffusion. We employ variational method for its error analysis and prove an improved convergence order of $$\frac 32$$ for the approximates of the solution. The core of the analysis is Hölder continuity in time and moment bounds for the solutions of the continuous and the discrete problem. Computational experiments are also presented. 

This paper is concerned with fully discrete finite element approximations of a stochastic nonlinear Schrödinger (sNLS) equation with linear multiplicative noise of the Stratonovich type. The goal of studying the sNLS equation is to understand the role played by the noises for a possible delay or prevention of the collapsing and/or blow-up of the solution to the sNLS equation. In the paper we first carry out a detailed analysis of the properties of the solution which lays down a theoretical foundation and guidance for numerical analysis, we then present a family of three-parameters fully discrete finite element methods which differ mainly in their time discretizations and contains many well-known schemes (such as the explicit and implicit Euler schemes and the Crank-Nicolson scheme) with different combinations of time discetization strategies. The prototypical \begin{document}$$ \theta $$\end{document}-schemes are analyzed in detail and various stability properties are established for its numerical solution. An extensive numerical study and performance comparison are also presented for the proposed fully discrete finite element schemes.