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We probe both the unidimensional quartic harmonic oscillator and the double well potential through a numerical analysis of the Functional Renormalization Group flow equations truncated at first order in the derivative expansion. The two partial differential equations for the potential V k (φ) and the wave function renormalization Z k (φ) , as obtained in different schemes and with distinct regulators, are studied down to k = 0 , and the energy gap between lowest and first excited state is computed, in order to test the reliability of the approach in a strongly non-perturbative regime. Our findings point out at least three ranges of the quartic coupling λ , one with higher λ where the lowest order approximation is already accurate, the intermediate one where the inclusion of the first correction produces a good agreement with the exact results and, finally, the one with smallest λ where presumably the higher order correction of the flow is needed. Some details of the specifics of the infrared regulator are also discussed. • A new numerical approach to non-perturbative computation in QFT. • Precise determination of tunneling rates • Numerical results comparable with instanton calculations and pertubation theory [ABSTRACT FROM AUTHOR]