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The explicit Euler scheme and similar explicit approximation schemes (such asthe Milstein scheme) are known to diverge strongly and numerically weakly inthe case of one-dimensional stochastic ordinary differential equations withsuperlinearly growing nonlinearities. It remained an open question whether sucha divergence phenomenon also holds in the case of stochastic partialdifferential equations with superlinearly growing nonlinearities such asstochastic Allen-Cahn equations. In this work we solve this problem by provingthat full-discrete exponential Euler and full-discrete linear-implicit Eulerapproximations diverge strongly and numerically weakly in the case ofstochastic Allen-Cahn equations. This article also contains a short literatureoverview on existing numerical approximation results for stochasticdifferential equations with superlinearly growing nonlinearities.

Strong and weak divergence of exponential and linear-implicit Euler  approximations for stochastic partial differential equations with  superlinearly growing nonlinearities