Measurement of the higher-order anisotropic flow coefficients for identified hadrons in Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV

New PHENIX measurements of the anisotropic flow coefficients $v_2\{{\mathrm{\Psi }}_2\}$, $v_3\{{\mathrm{\Psi }}_3\}$, $v_4\{{\mathrm{\Psi }}_4\}$ and $v_4\{{\mathrm{\Psi }}_2\}$ for identified particles (${\pi }^{\pm }$, $K^{\pm }$, and $p+\bar{p}$) obtained relative to the event planes ${\mathrm{\Psi }}_n$ in Au$+$Au collisions at $\sqrt{s_{{}_{NN}}}$ = 200 GeV are presented as functions of collision centrality and particle transverse momenta $p_T$. The $v_n$ coefficients show characteristic patterns consistent with hydrodynamical expansion of the matter produced in the collisions. For each harmonic $n$, a modified valence quark number $n_q$ scaling plotting $v_n\mathrm{/}(n_q{)}^{n\mathrm{/}2}$ versus ${\mathrm{K}\mathrm{E}}_T\mathrm{/}{n}_q$ is observed to yield a single curve for all the measured particle species for a broad range of transverse kinetic energies ${\mathrm{K}\mathrm{E}}_T$. A simultaneous blast wave model fit to the observed particle spectra and $v_n(p_T)$ coefficients identifies spatial eccentricities $s_n$ at freeze-out, which are much smaller than the initial-state geometric values.