We propose and analyse a boundary-preserving numerical scheme for the weak approximation for some stochastic partial differential equations (SPDEs) with bounded state-space. We impose regularity assumptions on the drift and diffusion coefficients only locally on the state-space. In particular, the drift and diffusion coefficients may be non-globally Lipschitz continuous and superlinearly growing. The scheme consists of a finite difference discretisation in space and a Lie--Trotter time splitting followed by exact simulation and exact integration in time. The proposed scheme converges in the weak sense of order $1/4$ in time and of order $1/2$ in space, for globally Lipschitz continuous test functions. We prove the weak convergence order in time by proving strong convergence towards a strong solution driven by a different noise process. The convergence order in space follows from known results. The boundary-preserving property is ensured by the use of Lie--Trotter time splitting followed by exact simulation and exact integration. Numerical experiments confirm the theoretical results and demonstrate the practical advantages of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing schemes for SPDEs.

翻译：本文针对具有有界状态空间的随机偏微分方程，提出并分析了一种保持边界的弱逼近数值格式。我们仅要求在状态空间局部上漂移系数和扩散系数满足正则性假设。特别地，漂移系数和扩散系数可以是非全局Lipschitz连续且超线性增长的。该格式包含空间上的有限差分离散化，以及Lie--Trotter时间分裂后接精确模拟和精确时间积分。对于全局Lipschitz连续的测试函数，所提格式在时间上以$1/4$阶、在空间上以$1/2$阶弱收敛。我们通过证明格式强收敛于一个由不同噪声过程驱动的强解，来证明时间方向上的弱收敛阶。空间方向的收敛阶则来自已知结果。边界保持特性通过采用Lie--Trotter时间分裂后接精确模拟和精确积分来保证。数值实验验证了理论结果，并证明了所提出的Lie--Trotter-精确（LTE）格式相较于现有SPDE格式的实际优势。