We study the numerical approximation of the stochastic heat equation with a distributional reaction term. Under a condition on the Besov regularity of the reaction term, it was proven recently that a strong solution exists and is unique in the pathwise sense, in a class of H\"older continuous processes. For a suitable choice of sequence $(b^k)_{k\in \mathbb{N}}$ approximating $b$, we prove that the error between the solution $u$ of the SPDE with reaction term $b$ and its tamed Euler finite-difference scheme with mollified drift $b^k$, converges to $0$ in $L^m(\Omega)$ with a rate that depends on the Besov regularity of $b$. In particular, one can consider two interesting cases: first, even when $b$ is only a (finite) measure, a rate of convergence is obtained. On the other hand, when $b$ is a bounded measurable function, the (almost) optimal rate of convergence $(\frac{1}{2}-\varepsilon)$-in space and $(\frac{1}{4}-\varepsilon)$-in time is achieved. Stochastic sewing techniques are used in the proofs, in particular to deduce new regularising properties of the discrete Ornstein-Uhlenbeck process.

翻译：我们研究了带分布形式反应项的随机热方程的数值逼近问题。在反应项满足贝索夫正则性条件下，最近有研究证明了在赫尔德连续过程类中，强解存在且在路径意义下唯一。对于逼近b的合适序列$(b^k)_{k\in \mathbb{N}}$，我们证明了具有反应项b的随机偏微分方程的解u与其采用平滑漂移$b^k$的驯服欧拉有限差分格式之间的误差，在$L^m(\Omega)$中收敛到0，且收敛速率取决于b的贝索夫正则性。特别地，我们可以考虑两个有趣的情况：首先，即使b仅为（有限）测度，也能获得收敛速率。另一方面，当b为有界可测函数时，实现了（几乎）最优收敛速率——空间方向$(\frac{1}{2}-\varepsilon)$和时间方向$(\frac{1}{4}-\varepsilon)$。证明中使用了随机缝补技术，特别用于推导离散奥恩斯坦-乌伦贝克过程的新正则化性质。