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[1] A discrete fracture model for the flow of compressible, multicomponent fluids in homogeneous, heterogeneous, and fractured media is presented in single phase. In the numerical model we combine the mixed finite element (MFE) and the discontinuous Galerkin (DG) methods. We use the cross-flow equilibrium concept to approximate the fractured matrix mass transfer. The discrete fracture model is numerically superior to the single-porosity model and overcomes limitations of the dual-porosity models including the use of a shape factor. The MFE method provides a direct and accurate approximation for the velocity field, which is crucial for the convective terms in the flow equations. The DG method associated with a slope limiter is used to approximate the species balance equations. This method can capture the sharp moving fronts. The calculation of the fracture-fracture flux across three and higher intersecting fracture branches is a challenge. In this work, we provide an accurate approximation of these fluxes by using the MFE formulation. Numerical examples in unfractured and fractured media illustrate the efficiency and robustness of the proposed numerical model.

SUMMARYIn this paper, we introduce an extension of Van Leer's slope limiter for two-dimensional discontinuous Galerkin (DG) methods on arbitrary unstructured quadrangular or triangular grids. The aim is to construct a non-oscillatory shock capturing DG method for the approximation of hyperbolic conservative laws without adding excessive numerical dispersion. Unlike some splitting techniques that are limited to linear approximations on rectangular grids, in this work, the solution is approximated by means of piecewise quadratic functions. The main idea of this new reconstructing and limiting technique follows a well-known approach where local maximum principle regions are defined by enforcing some constraints on the reconstruction of the solution. Numerical comparisons with some existing slope limiters on structured as well as on unstructured meshes show a superior accuracy of our proposed slope limiters.