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In this paper, we prove convergence for contractive time discretisationschemes for semi-linear stochastic evolution equations with irregular Lipschitznonlinearities, initial values, and additive or multiplicative Gaussian noiseon $2$-smooth Banach spaces $X$. The leading operator $A$ is assumed togenerate a strongly continuous semigroup $S$ on $X$, and the focus is onnon-parabolic problems. The main result concerns convergence of the uniformstrong error $$E_{k}^{\infty} := \Big(\mathbb{E} \sup_{j\in \{0, \ldots, N_k\}}\|U(t_j) - U^j\|_X^p\Big)^{1/p} \to 0\quad (k \to 0),$$ where $p \in[2,\infty)$, $U$ is the mild solution, $U^j$ is obtained from a timediscretisation scheme, $k$ is the step size, and $N_k = T/k$ for final time$T>0$. This generalises previous results to a larger class of admissiblenonlinearities and noise, as well as rough initial data from the Hilbert spacecase to more general spaces. We present a proof based on a regularisationargument. Within this scope, we extend previous quantified convergence resultsfor more regular nonlinearity and noise from Hilbert to $2$-smooth Banachspaces. The uniform strong error cannot be estimated in terms of the simplerpointwise strong error $$E_k := \bigg(\sup_{j\in \{0,\ldots,N_k\}}\mathbb{E}\|U(t_j) - U^{j}\|_X^p\bigg)^{1/p},$$ which most of the existing literature isconcerned with. Our results are illustrated for a variant of the Schr\"odingerequation, for which previous convergence results were not applicable.

Temporal approximation of stochastic evolution equations with irregular nonlinearities