We propose and analyse a boundary-preserving numerical scheme for the weak approximations of 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 splitting followed by exact simulation and exact integration in time. We prove weak convergence of optimal order 1/4 for globally Lipschitz continuous test functions of the scheme by proving strong convergence towards a strong solution driven by a different noise process. Boundary-preservation 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 effectiveness of the proposed Lie--Trotter-Exact (LTE) scheme compared to existing methods for SPDEs.

翻译：针对具有有界状态空间的一类随机偏微分方程（SPDEs）的弱近似问题，本文提出并分析了一种边界保持型数值格式。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，这些系数可以是非全局Lipschitz连续且具有超线性增长特性的。该数值格式包含空间方向的有限差分离散化，以及时间方向上采用Lie--Trotter分裂法结合精确模拟与精确积分的过程。通过证明格式在由不同噪声过程驱动的强解意义下的强收敛性，我们验证了该格式对于全局Lipschitz连续检验函数具有最优阶1/4的弱收敛性。边界保持特性通过Lie--Trotter时间分裂结合精确模拟与精确积分的方法得以保证。数值实验验证了理论结果，并证明了所提出的Lie--Trotter-Exact（LTE）格式相较于现有SPDEs求解方法的优越性。