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.

翻译：我们研究了粒子方法的计算能力，这类算法在科学计算与计算机仿真中有广泛应用。计算模型的计算能力决定了它能求解的问题类别。自动机理论可用于描述抽象机器（自动机）的计算能力及其可解问题。在乔姆斯基层级的形式语言与文法体系顶端是图灵机，它近似于现代计算机的构建原理。尽管粒子方法可根据其形式化定义被解释为自动机，但其计算能力迄今尚未被系统研究。本文通过分析粒子方法的图灵完备性来填补这一空白。具体而言，我们证明了在两组约束条件下粒子方法仍保持图灵完备性，并揭示了其丧失图灵完备性的情形。这一研究有助于理解粒子方法的理论基础，并为计算机仿真的功能边界提供洞见。