  We compare accuracy of two prime time evolution algorithms involving Matrix
Product States - tDMRG (time-dependent density matrix renormalization group)
and TDVP (time-dependent variational principle). The latter is supposed to be
superior within a limited and fixed auxiliary space dimension. Surprisingly, we
find that the performance of algorithms depends on the model considered. In
particular, many-body localized systems as well as the crossover regions
between localized and delocalized phases are better described by tDMRG,
contrary to the delocalized regime where TDVP indeed outperforms tDMRG in terms
of accuracy and reliability. As an example, we study many-body localization
transition in a large size Heisenberg chain. We discuss drawbacks of previous
estimates [Phys. Rev. B 98, 174202 (2018)] of the critical disorder strength
for large systems.
