A high order hybridizable discontinuous Galerkin method for incompressible miscible displacement in heterogeneous media

Abstract

We present a new method for approximating solutions to the incompressible miscible displacement problem in porous media. At the discrete level, the coupled nonlinear system has been split into two linear systems that are solved sequentially. The method is based on a hybridizable discontinuous Galerkin method for the Darcy flow, which produces a mass-conservative flux approximation, and a hybridizable discontinuous Galerkin method for the transport equation. The resulting method is high order accurate. Due to the implicit treatment of the system of partial differential equations, we observe computationally that no slope limiters are needed. Numerical experiments are provided that show that the method converges optimally and is robust for highly heterogeneous porous media in 2D and 3D.

High‐order hybridizable discontinuous Galerkin formulation with fully implicit temporal schemes for the simulation of two‐phase flow through porous media

“…11 Recently, the HDG method has been applied in two-phase flow through porous media. [18][19][20] It has all the advantages of the discontinuous Galerkin formulations. This method introduces the trace of the scalar variable as a new unknown.…”

“…2,[8][9][10][11][12][13][14][15][16][17] Recently, many efforts have been focused on applying high-order methods to these kind of problems due to their advantages. 11,15,[18][19][20] If the analytical solution is smooth enough, then the numerical solution obtained with a method of order k converges to the analytical one as h k e in L 2 -norm, being h e the element size of the mesh. [21][22][23] Hence, it has been shown that high-order spatial discretization methods can be more accurate than low-order ones for the same mesh resolution.…”

“…For these reasons, high-order HDG formulation has been recently applied in porous media flow problems. [17][18][19][20] To exploit the advantages of the high-order HDG formulation, we propose to perform a high-order fully implicit RK method to control the temporal error. These RK schemes are unconditionally stable for any combination of element size, polynomial degree and time-step.…”

Improved error estimates of hybridizable interior penalty methods using a variable penalty for highly anisotropic diffusion problems

“…Hybridizable discontinuous Galerkin (HDG) methods were first introduced in the last decade by Cockburn et al [1] (see, e.g., [2]) and have since received extensive attention from the research community. They are popular and very efficient numerical approaches for solving a large class of partial differential equations (see, e.g., [3,4,5,6,7] for a historical perspective). Indeed, they inherit attractive features from both (i) discontinuous Galerkin (DG) methods such as local conservation, hp-adaptivity and high-order polynomial approximation [8] and (ii) standard conforming Galerkin (CG) methods such as the Schur complement strategy [9].…”

Métodos de elementos finitos híbridos estáveis e estabilizados para escoamentos miscíveis em meios porosos heterogêneos

“…Adotando uma combinação de métodos de elementos finitos mistos clássicos com métodos DG, Yang e Chen [59] apresentam uma estratégia de pósprocessamento que resulta em uma superconvergência para a concentração, aumentando a precisão dos resultados da simulação. Métodos híbridos aplicados ao problema acoplado Darcy-Transporte [60,9,61,62,10] apresentam bons resultados. Seja combinados a outras estratégias, seja aproximando ambos os sistemas, sua formulação fica condicionada a imposições do problema modelado, como compatibilização dos espaços de aproximação em problemas mistos ou estabilização dos efeitos convectivos, quando necessário.…”