Bound-Preserving Discontinuous Galerkin Method for Compressible Miscible Displacement in Porous Media

Abstract

In this paper, we develop bound-preserving discontinuous Galerkin (DG) methods for the coupled system of compressible miscible displacement problems. We consider the problem with two components and the (volumetric) concentration of the ith component of the fluid mixture, c i , should be between 0 and 1. However, c i does not satisfy the maximum principle. Therefore, the numerical techniques introduced in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both c 1 and c 2 , respectively and enforce c 1 + c 2 = 1 simultaneously to obtain physically relevant approximations. By doing so, we have to treat the time derivative of the pressure dp/dt as a source in the concentration equation. Moreover, c ′ i s are not the conservative variables, as a result, the classical bound-preserving limiter in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 3091-3120) cannot be applied. Therefore, another limiter will be introduced. Numerical experiments will be given to demonstrate the accuracy in L ∞ -norm and good performance of the numerical technique.

High-Order Bound-Preserving Discontinuous Galerkin Methods for Stiff Multispecies Detonation

“…Recently, one of the authors studied miscible displacements in porous media and constructed a second-order DG scheme that preserves the two bounds 0 and 1 for the volumetric percentage in [5] on rectangular meshes, and the extension to triangular meshes has been given in [1]. In this paper, we follow the ideas given in [5,1] to gaseous detonation to construct high-order DG schemes on general rectangular and triangular meshes. The basic idea is to apply the PP technique to each r i (or z i ) and enforce M i=1 r i = ρ (or M i=1 z i = 1) by choosing consistent fluxes (see the definition 3.1).…”

“…The second difficulty is the construction of high-order time integration for the stiff source term. The time discretization in the analysis in [18,5,1] was chosen as Euler forward method. However, in gaseous detonation, k r (T ) would be a large constant, leading to an extremely stiff source s i .…”

“…However, in gaseous detonation, k r (T ) would be a large constant, leading to an extremely stiff source s i . Therefore, by applying the idea in [18,5,1], the time step will be significantly limited. One alternative is to consider backward Euler discretization and derive the PP technique.…”

“…To the best knowledge of the author, the only work in this direction is given in [12], where the maximum-principle-preserving technique was investigated for hyperbolic equations. However, by using backward Euler method, the scheme is only first-order accurate in time and the idea cannot be extended to high-order methods following [18,5,1] since no high-order SSP RK methods can be written as a convex combination of backward Euler methods [4]. Moreover, due to the time step restriction by the PP technique, any time integration that is the combination of Euler forward and backward Euler, such as Crank-Nicolson method, cannot be applied.…”

“…In this paper, we will modify the scheme introduced in [6] to preserve the total mass. Then we can apply the ideas in [5,1] to construct the bound-preserving technique. Since the time step is not too small, it is possible to sufficiently refine the mesh to capture the correct position of the shocks.…”