  In this paper, we develop discontinuous Galerkin (DG) methods for the
Ostrovsky-Vakhnenko (OV) equation, which yields the shock solutions and
singular soliton solutions, such as peakon, cuspon and loop solitons. The OV
equation has also been shown to have a bi-Hamiltonian structure. We directly
develop the energy stable or Hamiltonian conservative discontinuous Galerkin
(DG) schemes for the OV equation. Error estimates for the two energy stable
schemes are also proved. For some singular solutions, including cuspon and loop
soliton solutions, the hodograph transformation is adopted to transform the OV
equation or the generalized OV system to the coupled dispersionless (CD)
system. Subsequently, two DG schemes are constructed for the transformed CD
system. Numerical experiments are provided to demonstrate the accuracy and
capability of the DG schemes, including shock solution and, peakon, cuspon and
loop soliton solutions.
