1 research outputs found
Strong and weak divergence of exponential and linear-implicit Euler approximations for stochastic partial differential equations with superlinearly growing nonlinearities
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.Comment: 65 page