  Numerical methods for the Euler equations with a singular source are
discussed in this paper.
  The stationary discontinuity induced by the singular source and its coupling
with the convection of fluid presents challenges to numerical methods.
  We show that the splitting scheme is not well-balanced and leads to incorrect
results; in addition, some popular well-balanced schemes also give incorrect
solutions in extreme cases due to the singularity of source.
  To fix such difficulties, we propose a solution-structure based approximate
Riemann solver, in which the structure of Riemann solution is first predicted
and then its corresponding approximate solver is given.
  The proposed solver can be applied to the calculation of numerical fluxes in
a general finite volume method, which can lead to a new well-balanced scheme.
  Numerical tests show that the discontinuous Galerkin method based on the
present approximate Riemann solver has the ability to capture each wave
accurately.
