We present high-order numerical schemes for linear stochastic heat and wave equations with Dirichlet boundary conditions, driven by additive noise. Standard Euler schemes for SPDEs are limited to an order convergence between 1/2 and 1 due to the low temporal regularity of noise. For the stochastic heat equation, a modified Crank-Nicolson scheme with proper numerical quadrature rule for the noise term in its reformulation as random PDE achieves a strong convergence rate of 3/2. For the stochastic wave equation with additive noise a corresponding approach leads to a scheme which is of order 2.

翻译：本文针对具有Dirichlet边界条件、由加性噪声驱动的线性随机热方程和波动方程，提出了高阶数值格式。由于噪声的时间正则性较低，SPDE的标准欧拉格式收敛阶被限制在1/2至1之间。对于随机热方程，通过在其重构为随机PDE时对噪声项采用适当的数值积分规则，改进的Crank-Nicolson格式实现了3/2阶的强收敛速率。对于具有加性噪声的随机波动方程，相应方法导出的数值格式具有2阶收敛精度。