Research on the accuracy of the discontinuous Galerkin method in the calculation of solutions with shock waves

The accuracy of the discontinuous Galerkin method of higher-order accuracy on smooth solutions is studied. Calculations were made for discontinuous solutions for a quasilinear hyperbolic system of conservation laws with shock waves propagating with a variable speed. As an example an approximation of the system of conservation laws of the theory of shallow water equations was chosen. It was shown that the discontinuous Galerkin method, in spite of high accuracy on smooth solutions and localization of shock waves, reduces its order of convergence to the first order in the areas of influence of shock waves.