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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. 

Abstract The numerical analysis of stochastic parabolic partial differential equations of the form$$\begin{aligned} du + A(u)\, dt = f \,dt + g \, dW, \end{aligned}$$ $\begin{array}{c}du+A\left(u\right)dt=fdt+gdW,\end{array}$is surveyed, whereAis a nonlinear partial operator andWa Brownian motion. This manuscript unifies much of the theory developed over the last decade into a cohesive framework which integrates techniques for the approximation of deterministic partial differential equations with methods for the approximation of stochastic ordinary differential equations. The manuscript is intended to be accessible to audiences versed in either of these disciplines, and examples are presented to illustrate the applicability of the theory. 

Abstract We propose some new mixed finite element methods for the time-dependent stochastic Stokes equations with multiplicative noise, which use the Helmholtz decomposition of the driving multiplicative noise. It is known (Langa, J. A., Real, J. & Simon, J. (2003) Existence and regularity of the pressure for the stochastic Navier--Stokes equations. Appl. Math. Optim., 48, 195--210) that the pressure solution has low regularity, which manifests in suboptimal convergence rates for well-known inf-sup stable mixed finite element methods in numerical simulations; see Feng X., & Qiu, H. (Analysis of fully discrete mixed finite element methods for time-dependent stochastic Stokes equations with multiplicative noise. arXiv:1905.03289v2 [math.NA]). We show that eliminating this gradient part from the noise in the numerical scheme leads to optimally convergent mixed finite element methods and that this conceptual idea may be used to retool numerical methods that are well known in the deterministic setting, including pressure stabilization methods, so that their optimal convergence properties can still be maintained in the stochastic setting. Computational experiments are also provided to validate the theoretical results and to illustrate the conceptual usefulness of the proposed numerical approach.