We construct a positivity-preserving Lie--Trotter splitting scheme with finite difference discretization in space for approximating the solutions to a class of nonlinear stochastic heat equations with multiplicative space-time white noise. We prove that this explicit numerical scheme converges in the mean-square sense, with rate $1/4$ in time and rate $1/2$ in space, under appropriate CFL conditions. Numerical experiments illustrate the superiority of the proposed numerical scheme compared with standard numerical methods which do not preserve positivity.

翻译：我们构建了一个结合空间有限差分离散的正性保持Lie-Trotter分裂格式，用于逼近一类带有乘性时空白噪声的非线性随机热方程的解。我们证明，在适当的CFL条件下，该显式数值格式在均方意义下收敛，时间方向收敛阶为$1/4$，空间方向收敛阶为$1/2$。数值实验表明，与不保持正性的标准数值方法相比，所提出的数值格式具有优越性。