Shock Capturing with Discontinuous Galerkin Method

Shock capturing has been a challenge for computational fluid dynamicists over the years. This article deals with discontinuous Galerkin method to solve the hyperbolic equations in which solutions may develop discontinuities in finite time. The high order discontinuous Galerkin method combining the basis of finite volume and finite element methods has shown a lot of attractive features for a wide range of applications. Various techniques proposed in the literature to deal with discontinuities basically reduce the order of interpolation in the region around these discontinuities. The accuracy of the scheme therefore may be degraded in the vicinity of the shock. The proposed method resolves the discontinuities presented in the solution by applying viscosity into the shock-containing elements. The discontinuity is spread over a distance and is well approximated in the space of interpolation functions. The technique of adding viscosity to the system and the indicator based on the expansion coefficients of the solution are presented. A number of numerical examples in one and two dimensions is carried out to show the capability of the scheme for shock capturing.