TY - JOUR
T1 - A Third Order Bound-Preserving Nodal Discontinuous Galerkin Method for Miscible Displacements in Porous Media
AU - Liu , Xinyuan
AU - Xiong , Tao
AU - Yang , Yang
JO - Communications in Computational Physics
VL - 5
SP - 1515
EP - 1551
PY - 2025
DA - 2025/09
SN - 38
DO - http://doi.org/10.4208/cicp.OA-2023-0046
UR - https://global-sci.org/intro/article_detail/cicp/24465.html
KW - Miscible displacements, bound-preserving, nodal discontinuous Galerkin method, multi-component fluid.
AB - <p style="text-align: justify;">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.</p>