We propose a modification of the standard linear implicit Euler integrator for the weak approximation of parabolic semilinear stochastic PDEs driven by additive space-time white noise. This new method can easily be combined with a finite difference method for the spatial discretization. The proposed method is shown to have improved qualitative properties compared with the standard method. First, for any time-step size, the spatial regularity of the solution is preserved, at all times. Second, the proposed method preserves the Gaussian invariant distribution of the infinite dimensional Ornstein–Uhlenbeck process obtained when the nonlinearity is removed, for any time-step size. The weak order of convergence of the proposed method is shown to be equal to 1/2 in a general setting, like for the standard Euler scheme. A stronger weak approximation result is obtained when considering the approximation of a Gibbs invariant distribution, when the nonlinearity is a gradient: one obtains an approximation in total variation distance of order 1/2, which does not hold for the standard method. This is the first result of this type in the literature and this is the major and most original result of this article.

我们提出了对标准线性隐式欧拉积分器的修改，用于由加性时空白噪声驱动的抛物线半线性随机偏微分方程的弱近似。这种新方法可以很容易地与空间离散化的有限差分方法结合起来。与标准方法相比，所提出的方法具有改进的定性特性。首先，对于任何时间步长，解的空间规律性始终保持不变。其次，对于任何时间步长，所提出的方法都保留了消除非线性时获得的无限维 Ornstein-Uhlenbeck 过程的高斯不变分布。在一般设置中，所提出方法的弱收敛阶数等于 1/2，就像标准欧拉方案一样。当考虑吉布斯不变分布的近似时，当非线性是梯度时，获得更强的弱近似结果：获得1/2阶总变差距离的近似，这对于标准方法不成立。这是文献中第一个此类结果，也是本文的主要且最具原创性的结果。