  Spectra of the neutral heavy mesons, $\eta_c(1S,2S)$, $J/\psi$, $\psi(2S)$,
$\eta_b(1S,2S,3S)$, $\Upsilon(1S,2S,3S)$, $D$, $D^\ast$, $B$, $B^\ast$, $B_s$
and $B_s^\ast$, in a homogeneous magnetic field are analyzed in a potential
model of constituent quarks. To obtain anisotropic wave functions and the
corresponding eigenvalues, the cylindrical Gaussian expansion method is
applied, where the wave functions for transverse and longitudinal directions in
the cylindrical coordinate are expanded by the Gaussian bases separately.
Energy level structures in the wide range of magnetic fields are obtained and
the deformation of the wave functions is shown, which reflects effects of the
spin mixing, the Zeeman splitting and quark Landau levels. The contribution
from the magnetic catalysis in heavy-light mesons is discussed as a change of
the light constituent quark mass.
