  The explicit Euler scheme and similar explicit approximation schemes (such as
the Milstein scheme) are known to diverge strongly and numerically weakly in
the case of one-dimensional stochastic ordinary differential equations with
superlinearly growing nonlinearities. It remained an open question whether such
a divergence phenomenon also holds in the case of stochastic partial
differential equations with superlinearly growing nonlinearities such as
stochastic Allen-Cahn equations. In this work we solve this problem by proving
that full-discrete exponential Euler and full-discrete linear-implicit Euler
approximations diverge strongly and numerically weakly in the case of
stochastic Allen-Cahn equations. This article also contains a short literature
overview on existing numerical approximation results for stochastic
differential equations with superlinearly growing nonlinearities.
