A Third Order Bound-Preserving Nodal Discontinuous Galerkin Method for Miscible Displacements in Porous Media

Abstract

In this paper, we develop a provable third order bound-preserving (BP) nodal discontinuous Galerkin (DG) method for compressible miscible displacements. We consider the problem with a multi-component fluid mixture and physically the volumetric concentration of each component, $c_j(j=1,···,N)$, is between 0 and 1. The main idea is to apply a positivity-preserving (PP) method to all $c′_js,$ while enforce $∑_jc_j=1$ simultaneously. First, we treat the time derivative of the pressure as a source and choose suitable “consistent” numerical fluxes in the pressure and concentration equations to construct a nodal interior penalty DG (IPDG) method to enforce $∑_jc_j =1.$ For PP, we represent the cell average of $c_j$ as a weighted summation of Gaussian quadrature point values, and transform which to some other specially chosen point values. We prove that by taking appropriate parameters in the nodal IPDG method and a suitable time stability condition, the cell average can be kept positive, which further implies that the cell averages of all components are between 0 and 1. Finally, we apply a polynomial scaling limiter to obtain physically relevant numerical approximations without sacrificing accuracy. Numerical experiments are given to demonstrate desired accuracy, BP and good performances of our proposed approach.