  We investigate the computational power of particle methods, a
well-established class of algorithms with applications in scientific computing
and computer simulation. The computational power of a compute model determines
the class of problems it can solve. Automata theory allows describing the
computational power of abstract machines (automata) and the problems they can
solve. At the top of the Chomsky hierarchy of formal languages and grammars are
Turing machines, which resemble the concept on which most modern computers are
built. Although particle methods can be interpreted as automata based on their
formal definition, their computational power has so far not been studied. We
address this by analyzing Turing completeness of particle methods. In
particular, we prove two sets of restrictions under which a particle method is
still Turing powerful, and we show when it loses Turing powerfulness. This
contributes to understanding the theoretical foundations of particle methods
and provides insight into the powerfulness of computer simulations.
