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We study relaxation processes in spin systems near criticality after a quenchfrom a high-temperature initial state. Special attention is paid to the stagewhere universal behavior, with increasing order parameter emerges from an earlynon-universal period. We compare various algorithms, lattice types, andupdating schemes and find in each case the same universal behavior atmacroscopic times, despite of surprising differences during the earlynon-universal stages.Comment: 9 pages, 3 figures, RevTeX, submitted to Phys. Rev. Let

Microscopic nonuniversality versus macroscopic universality in algorithms for critical dynamics