  We derive flow equations for cold atomic gases with one macroscopically
populated energy level. The generator is chosen such that the ground state
decouples from all other states in the system as the renormalization group flow
progresses. We propose a self-consistent truncation scheme for the flow
equations at the level of three-body operators and show how they can be used to
calculate the ground state energy of a general $N$-body system. Moreover, we
provide a general method to estimate the truncation error in the calculated
energies. Finally, we test our scheme by benchmarking to the exactly solvable
Lieb-Liniger model and find good agreement for weak and moderate interaction
strengths.
