  We present the numerical flow iteration (NuFI) for solving the
Vlasov--Poisson equation. In a certain sense specified later herein, NuFI
provides infinite resolution of the distribution function. NuFI exactly
preserves positivity, all $L^p$-norms, charge, and entropy. Numerical
experiments show no energy drift. NuFI is fast, requires several orders of
magnitude less memory than conventional approaches, and can very efficiently be
parallelised on GPU clusters. Low fidelity simulations provide good qualitative
results for extended periods of time and can be computed on low-cost
workstations.
