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：本文研究了一类带乘性噪声的半线性随机偏微分方程的时间半离散与空间全离散的新方法。方程漂移项仅需满足单边Lipschitz条件，扩散项则假设为全局Lipschitz连续。我们提出的时间离散新策略基于随机微分方程中的Milstein方法，利用能量法进行误差分析，并证明近似解具有接近$1$阶的强收敛阶。该证明依赖于SPDE解和非线性项的新Hölder连续性估计。对于一般多项式型漂移项，即使数值解的稳定性也难以直接推导。为克服这一困难，我们提出基于插值的有限元方法进行空间离散。通过数值与随机技术，我们获得了$H^1$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。结合上述所有结果，进而得到时间与空间上的接近最优收敛阶。最后通过数值实验实现所提出的数值格式，并验证理论结果。