  In this work we develop several new simulation algorithms for 1D many-body
quantum mechanical systems combining the Matrix Product State variational
ansatz with Taylor, Pade and Arnoldi approximations to the evolution operator.
By comparing all methods with previous techniques based on Trotter
decompositions we demonstrate that the Arnoldi method is the best one, reaching
extremely good accuracy with moderate resources. Finally we apply this
algorithm to studying the formation of molecules in an optical lattices when
crossing a Feschbach resonance with a cloud of two-species hard-core bosons.
