  We consider numerical solutions for the Allen-Cahn equation with standard
double well potential and periodic boundary conditions. Surprisingly it is
found that using standard numerical discretizations with high precision
computational solutions may converge to completely incorrect steady states.
This happens for very smooth initial data and state-of-the-art algorithms. We
analyze this phenomenon and showcase the resolution of this problem by a new
symmetry-preserving filter technique. We develop a new theoretical framework
and rigorously prove the convergence to steady states for the filtered
solutions.
