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The blazar Mrk 501 (z=0.034) was observed at very-high-energy (VHE, $E\gtrsim100$~GeV) gamma-ray wavelengths during a bright flare on the night of 2014 June23-24 (MJD 56832) with the H.E.S.S. phase-II array of Cherenkov telescopes.Data taken that night by H.E.S.S. at large zenith angle reveal an exceptionalnumber of gamma-ray photons at multi-TeV energies, with rapid flux variabilityand an energy coverage extending significantly up to 20 TeV. This data set isused to constrain Lorentz invariance violation (LIV) using two independentchannels: a temporal approach considers the possibility of an energy dependencein the arrival time of gamma rays, whereas a spectral approach considers thepossibility of modifications to the interaction of VHE gamma rays withextragalactic background light (EBL) photons. The non-detection ofenergy-dependent time delays and the non-observation of deviations between themeasured spectrum and that of a supposed power-law intrinsic spectrum withstandard EBL attenuation are used independently to derive strong constraints onthe energy scale of LIV ($E_{\rm{QG}}$) in the subluminal scenario for linearand quadratic perturbations in the dispersion relation of photons. For the caseof linear perturbations, the 95% confidence level limits obtained are$E_{\rm{QG},1} > 3.6 \times 10^{17} \ \rm{GeV} $ using the temporal approachand $E_{\rm{QG},1} > 2.6 \times 10^{19} \ \rm{GeV}$ using the spectralapproach. For the case of quadratic perturbations, the limits obtained are$E_{\rm{QG},2} > 8.5 \times 10^{10} \ \rm{GeV} $ using the temporal approachand $E_{\rm{QG},2} > 7.8 \times 10^{11} \rm{ GeV}$ using the spectral approach.

The 2014 TeV Gamma-ray Flare of Mrk 501 Seen with H.E.S.S.: Temporal and Spectral Constraints on Lorentz Invariance Violation