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The functional renormalization group (FRG) approach is a powerful tool forstudies of a large variety of systems, ranging from statistical physics overthe theory of the strong interaction to gravity. The practical application ofthis approach relies on the derivation of so-called flow equations, whichdescribe the change of the quantum effective action under the variation of acoarse-graining parameter. In the present work, we discuss in detail a novelapproach to solve such flow equations. This approach relies on the fact that RGequations can be rewritten such that they exhibit similarities with theconservation laws of fluid dynamics. This observation can be exploited indifferent ways. First of all, we show that this allows to employ powerfulnumerical techniques developed in the context of fluid dynamics to solve RGequations. In particular, it allows us to reliably treat the emergence ofnonanalytic behavior in the RG flow of the effective action as it is expectedto occur in studies of, e.g., spontaneous symmetry breaking. Second, theanalogy between RG equations and fluid dynamics offers the opportunity to gainnovel insights into RG flows and their interpretation in general, including theirreversibility of RG flows. We work out this connection in practice byapplying it to zero-dimensional quantum-field theoretical models. Thegeneralization to higher-dimensional models is also discussed. Our findings areexpected to help improving future FRG studies of quantum field theories inhigher dimensions both on a qualitative and quantitative level.

Numerical fluid dynamics for FRG flow equations: Zero-dimensional QFTs as numerical test cases. I. The $O(N)$ model