Microscopic nonuniversality versus macroscopic universality in algorithms for critical dynamics

We study relaxation processes in spin systems near criticality after a quench from a high-temperature initial state. Special attention is paid to the stage where universal behavior, with increasing order parameter m(t){~}t^θ, emerges from an early nonuniversal period. We compare various algorithms, lattice types, and updating schemes and find in each case the same universal behavior at macroscopic times, despite surprising differences during the early nonuniversal stages.