Discontinuous Galerkin method in fluid dynamics problems with nonsmooth solutions

We discuss the way of using the discontinuous Galerkin method to solve fluid dynamics problems with nonsmooth solutions. A distinguishing feature of this approach is that the Zhmakin–Fursenko monotonic scheme, which is based on the elective diffusion–antidiffusion approach, is directly applied to the numerical representation of the solution. It enables us to preserve the logic of the discontinuous Galerkin method. The validity of the proposed approach is verified by solving the known test problems such as discontinuity disintegration (Riemann problem) and shock wave propagation through a variable medium (Shu–Osher problem).