In this paper, we consider a new approach for semi-discretization in time and spatial discretization of a class of semi-linear stochastic partial differential equations (SPDEs) with multiplicative noise. The drift term of the SPDEs is only assumed to satisfy a one-sided Lipschitz condition and the diffusion term is assumed to be globally Lipschitz continuous. Our new strategy for time discretization is based on the Milstein method from stochastic differential equations. We use the energy method for its error analysis and show a strong convergence order of nearly $1$ for the approximate solution. The proof is based on new H\"older continuity estimates of the SPDE solution and the nonlinear term. For the general polynomial-type drift term, there are difficulties in deriving even the stability of the numerical solutions. We propose an interpolation-based finite element method for spatial discretization to overcome the difficulties. Then we obtain $H^1$ stability, higher moment $H^1$ stability, $L^2$ stability, and higher moment $L^2$ stability results using numerical and stochastic techniques. The nearly optimal convergence orders in time and space are hence obtained by coupling all previous results. Numerical experiments are presented to implement the proposed numerical scheme and to validate the theoretical results.

翻译：本文针对一类具有乘性噪声的半线性随机偏微分方程（SPDEs），提出了一种新的时间半离散与空间离散方法。SPDEs的漂移项仅需满足单侧Lipschitz条件，扩散项假设为全局Lipschitz连续。我们的时间离散新策略基于随机微分方程中的Milstein方法。采用能量法进行误差分析，证明了近似解具有接近$1$的强收敛阶。该证明基于SPDE解及其非线性项新的Hölder连续性估计。对于一般多项式型漂移项，即使是数值解的稳定性推导也存在困难。为克服这一困难，我们提出了一种基于插值的有限元方法进行空间离散。随后，利用数值与随机技巧，获得了$H^1$稳定性、高阶矩$H^1$稳定性、$L^2$稳定性及高阶矩$L^2$稳定性结果。通过耦合前述所有结果，进而获得了时间与空间上的近似最优收敛阶。最后，通过数值实验实现了所提数值格式，并验证了理论结果。