Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

In this paper, we develop discontinuous Galerkin 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 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 discontinuous Galerkin schemes are constructed for the transformed CD system. Numerical experiments are provided to demonstrate the accuracy and capability of the proposed schemes, including shock solution and, peakon, cuspon and loop soliton solutions.