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（SPDEs），提出并分析了一种边界保持数值格式以实现弱逼近。我们仅要求在状态空间局部范围内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连续且具有超线性增长特性的。该数值格式包含空间上的有限差分离散化、Lie--Trotter时间分裂法，以及后续的精确模拟与时间精确积分。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阶的弱收敛速度收敛。我们通过证明该格式在强收敛意义下趋近于由不同噪声过程驱动的强解，从而验证了时间方向上的弱收敛阶。空间收敛阶则基于已有结果得出。边界保持特性通过采用Lie--Trotter时间分裂法结合精确模拟与精确积分得以保证。数值实验验证了理论结果，并展示了所提出的Lie--Trotter-精确（LTE）格式相较于现有SPDE数值格式的实用优势。