We use overlap fermions as valence quarks to calculate meson masses in a wide quark mass range on the $2+1$-flavor domain-wall fermion gauge configurations generated by the RBC and UKQCD Collaborations. The well-defined quark masses in the overlap fermion formalism and the clear valence quark mass dependence of meson masses observed from the calculation facilitate a direct derivation of physical current quark masses through a global fit to the lattice data, which incorporates $O(a^2)$ and $O(m_c^4a^4)$ corrections, chiral extrapolation, and quark mass interpolation. Using the physical masses of $D_s$, $D_s^*$ and $J/psi$ as inputs, Sommers scale parameter $r_0$ and the masses of charm quark and strange quark in the $overline{rm MS}$ scheme are determined to be $r_0=0.465(4)(9)$ fm, $m_c^{overline{rm MS}}(2,{rm GeV})=1.118(6)(24)$ GeV (or $m_c^{overline{rm MS}}(m_c)=1.304(5)(20)$ GeV), and $m_s^{overline{rm MS}}(2,{rm GeV})=0.101(3)(6),{rm GeV}$, respectively. Furthermore, we observe that the mass difference of the vector meson and the pseudoscalar meson with the same valence quark content is proportional to the reciprocal of the square root of the valence quark masses. The hyperfine splitting of charmonium, $M_{J/psi}-M_{eta_c}$, is determined to be 119(2)(7) MeV, which is in good agreement with the experimental value. We also predict the decay constant of $D_s$ to be $f_{D_s}=254(2)(4)$ MeV. The masses of charmonium $P$-wave states $chi_{c0}, chi_{c1}$ and $h_c$ are also in good agreement with experiments.
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