Bound-preserving discontinuous Galerkin method for compressible miscible displacement in porous media ∗

Bound-preserving discontinuous Galerkin method for compressible miscible displacement in porous media

Hui Guo^† Yang Yang^‡

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,cⁱ, should be between 0 and 1. However,cⁱdoes 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 bothc₁ andc₂, respectively and enforcec₁+c₂= 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^′is 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 inL^∞-norm and good performance of the numerical technique.

Keywords: Compressible miscible displacement problem; Discontinuous Galerkin method; Bound-preserving

1 Introduction

Numerical modeling of miscible displacement in porous media is important and interesting in oil recovery and environmental pollution problem. The miscible displacement problem is described by a coupled system of nonlinear partial differential equations. The need for accurate solutions to the coupled equations challenges numerical analysts to design new methods. In this paper, we study the compressible miscible displacement problem in porous media on computational domain Ω = [0,2π]× [0,2π]

d(c)p_t+∇ ·u=d(c)p_t− ∇ ·(^κ(x,y)_µ(c) ∇p) =q, (x, y)∈Ω, 0< t≤T,

φc_t+b(c)p_t+u· ∇c− ∇ ·(D∇c) = (˜c−c)q, (x, y)∈Ω, 0< t≤T, (1.1) as well as its one-dimensional version. In (1.1) the dependent variables p, u and c are the pressure in the fluid mixture, the Darcy velocity of the mixture (volume flowing across a unit across-section

∗Supported by National Natural Science Foundation of China (11571367 and 11601536) and Michigan Tech- nological University Research Excellence Fund Scholarship and Creativity Grant 1605052

†College of Science, China University of Petroleum, Qingdao 266580, China. E-mail: [email protected]

‡Michigan Technological University, Houghton, MI 49931, USA. E-mail:[email protected]

per unit time), and the concentration of interested species measured in amount of species per unit volume of the fluid mixture, respectively. φand κ are the porosity and the permeability of the rock, respectively. µis the concentration-dependent viscosity. q is the external volumetric flow rate, and ˜cis the concentration of the fluid in the external flow. ˜cmust be specified at points where injection (q >0) takes place, and is assumed to be equal to c at production points (q <0). The diffusion coefficient D arises from two aspects: molecular diffusion, which is rather small for field-scale problems, and dispersion, which is velocity-dependent, in the petroleum engineering literature. Its form is

D=

D₁₁ D₁₂ D₂₁ D₂₂

=φ(x, y)(d_molI+d_long|u|E+dtran|u|E^⊥), (1.2) where E, a 2×2 matrix, represents the orthogonal projection along the velocity vector and is given by

E= (e_ij(u)) = u_iu_j

|u|²

, u= (u₁, u₂),

and E^⊥=I−Eis the orthogonal complement. The diffusion coefficientd_long measures the dispersion in the direction of the flow and dtran shows that transverse to the flow. To ensure the stability of the scheme, in almost all of the previous worksDis assumed to be strictly positive definite. In this paper, we assume D to be positive semidefinite. Therefore, we have D₁₁ ≥ 0, D₂₂ ≥ 0 and D₁₂ =D₂₁. In the numerical experiments, we also choose D=0 to challenge the algorithm. Moreover, the pressure is uniquely determined up to a constant, thus we assume R

Ωpdxdy = 0 at t = 0. However, this assumption is not essential. In this paper, we consider a two component displacement only. The coefficients can be stated as follows:

c=c₁ = 1−c₂, a(c) =a(x, y, c) = ^µ(c)_κ , b(c) =b(x, y, c) =φc₁{z₁− P²

j=1

zjcj}, d(c) =d(x, y, c) =φP²

j=1

zjcj,

whereci andzi are the concentration and the compressibility factor of the ith component of the fluid mixture, respectively. In this problem, the Neumann boundary conditions are given as

u·n= 0, (D∇c−cu)·n= 0, (1.3)

where n is the outer normal of the boundary∂Ω. Moreover, the initial solutions are given as c(x, y,0) =c₀(x, y), p(x, y,0) =p₀(x, y), (x, y)∈Ω.

The miscible displacement problem in porous media was first introduced in [9, 10], in which the mixed finite element methods were presented. Later, the compressible problem was studied in [11]

and the optimal order estimates in L² and almost optimal order estimates in L^∞ were given in [5].

Subsequently, many new methods were introduced to solve the compressible miscible displacements, such as finite difference method [43, 44, 45], characteristics collocation method [21], splitting positive definite mixed element method [35] and H¹-Galerkin mixed method [3]. Besides the above, in [30], an accurate and efficient simulator is developed for problems with wells. Later, the authors introduced

an Eulerian-Lagrangian localized adjoint method (ELLAM) to solve the transport partial differen- tial equation for concentration, while a mixed finite element method (MFEM) to solve the pressure equation [29]. Recently, discontinuous Galerkin (DG) methods became more popular in solving com- pressible miscible displacement problem in porous media [6, 7, 36, 38, 15, 37]. Some special numerical techniques were introduced to control jumps of numerical approximations as well as the nonlinearality of the convection term. Moreover, the superconvergence results based on the DG methods were also proved in [39, 40]. Besides the above, there were also significant works discussing the discontinuous Galerkin methods for incompressible miscible displacements, see e.g. [1, 16, 19, 23, 26, 27, 31] and for general porous media flow, see e.g. [2, 13, 12, 28] and the references therein. However, to the best knowledge of the authors, no previous works focused on the bound-preserving techniques. In many numerical simulations, the approximations of c can be placed out of the interval [0,1]. Especially for problems with large gradients, the value of d(c) might be negative, leading to ill-posedness of the problem, and the numerical approximations will blow up. We will use numerical experiments to demonstrate this point in Section 4. In this paper, we will discuss the bound-preserving technique for compressible problems. However, the idea can be extended to incompressible flows with some minor changes.

The DG method gained even greater popularity for good stability, high order accuracy, and flexi- bility on h-p adaptivity and on complex geometry. Physically bound-preserving high-order numerical methods for conservation laws have been actively studied in the last few years. Most cases, the DG schemes were coupled with suitable trouble cell indicators to avoid the order deterioration in regions where the solutions are smooth while maintaining physical solutions near shocks, e.g. [8, 18]. In 2010, the genuinely maximum-principle-satisfying high order DG and finite volume schemes were con- structed in [46] by Zhang and Shu. Subsequently, this technique has been successfully extended to compressible Euler equations without or with source terms [47, 48], shallow water equations [32], and hyperbolic equations with δ-singularities [41]. The basic idea is to take the test function to be 1 in each cell to obtain an equation of the numerical cell average of the target variable, say r, and prove the cell average, ¯r, is within the desired bounds. Then we can apply the bound-preserving limiter to the numerical approximation and construct a new one

˜

r = ¯r+θ(r−r),¯ θ∈[0,1]. (1.4)

If the problem has only one lower bound zero, the technique is also called positivity-preserving tech- nique. Thanks to the limiter, the whole algorithm were proved to beL¹stable [42, 22] for some compli- cated systems. Moreover, the technique does not rely on the trouble cell detector and the limiter keeps the high-order accuracy in regions with smooth solutions for scalar equations [46]. For convection- diffusion equations, in [49], the authors applied the same idea to construct genuinely second-order maximum-principle-satisfying DG methods on unstructured meshes. Subsequently, the ideas have been extended to semilinear parabolic equations with blow-up solutions [17] and chemotaxis models [20] by using the LDG methods. Recently, the flux limiter [34, 33] and third-order maximum-preserving direct DG method [4] were also introduced. In this paper, we will extend the ideas in [46, 49] and

construct second-order bound-preserving DG methods. However, there are significant differences from previous techniques. First of all, most of the problems in [46, 49] satisfy maximum principles while the concentration c in (1.1) does not. Therefore, we will split the whole algorithm into two parts.

We first rewrite the system into a conservative form and treat pt as another source in the equation of concentration to obtain the positivity of cby the positivity-preserving technique [47, 48]. Then we choose a consistent flux pair (see Definition 2.1) in the concentration and pressure equations to obtain the positivity of 1−c. More precisely, in our analysis, instead of solving c and p, we rewrite (1.1) into a system of c and c₂= 1−cand enforce c+c₂ = 1 by choosing a consistent flux pair. Secondly, to apply the positivity-preserving technique, we need to numerically approximate the conservative variable r = φc instead of c. By doing so, the upper bound of r is not a constant and the limiter (1.4) may fail to work, since such a θ may not exist. Therefore, we will introduce another limiter to obtain physically relevant numerical approximations. Before we finish the introduction, we would like to point out the main difference between the proposed scheme and the TVD-like schemes. For TVD schemes, the numerical approximations are only first-order accurate at the local extrema. In this paper, our scheme is second-order accurate inL^∞-norm. In Section 4, we will construct analytical solutions to demonstrate the accuracy. To the best knowledge of the authors, this is the first analytical solution available.

The paper is organized as follows. In Sections 2 and 3, we present the bound-preserving technique for problems in one and two space dimensions. Analytical solutions will be constructed and some numerical results will be given to demonstrate the good performance of the bound-preserving DG method in Section 4. We will end in Section 5 with concluding remarks and remarks for future works.

2 Numerical algorithm in one space dimension

In this section, we proceed to construct the bound-preserving DG scheme to solve the one-dimensional compressible miscible displacement problem in porous media.

2.1 The DG scheme

We consider the one-dimensional version of (1.1) on the spatial domain Ω = [0,2π] and solve d(c)pt+ux=d(c)pt−(^k(x)_µ(c)px)x=q, x∈Ω, 0< t≤T,

φct+b(c)pt+ucx−(D(u)cx)x = (˜c−c)q, x∈Ω, 0< t≤T, with Neuamnn boundary conditions

u = 0, D(u)c_x−cu= 0 on ∂Ω. (2.1)

We first rewrite the system into a conservative form

d(c)pt+ux=q, (2.2)

a(c)u=−px, (2.3)

(φc)t+ (uc)x−(D(u)cx)x = ˜cq−φcz₁pt, (2.4)

where a(c) = ^µ(c)_κ and z₁ is the compressibility factor of first component of the fluid mixture. To construct the DG scheme, we divide the computational domain Ω into N cells

0 =x¹

2 < x³

2 <· · ·< x_N+¹

2 = 2π, and denote

Ij = x_j−¹

2, x_j+¹

2

, j= 1,· · · , N

as the cells. For simplicity, we consider uniform meshes in this paper, and denote by ∆x the size of each cell. However, this assumption is not essential. Following [49], we consider second-order DG scheme only, and define

V_h =

v∈L²(Ω) :v|^Ij ∈ P¹(Ij), j = 1,· · · , N

as the finite element space, where P¹(Ij) denotes the space of linear polynomials in Ij. Moreover, for any v ∈ V_h, we use v⁻

j+¹₂ =v(x⁻

j+¹₂) to denote the left limit ofv at x_j+¹

2. Likewise for v⁺. Then the jump and average of v at x_j+¹

2 are given as [v]_j+¹

2 =v⁺

j+¹₂ −v⁻

j−¹2

and {v}_j+¹₂ = ¹₂(v⁺

j+¹₂ +v⁻

j−¹2

), respectively. For simplicity, we definev⁻1

2

=v⁺

N+¹₂ = 0. Before constructing the bound-preserving DG scheme, we would like to demonstrate the following main points that are quite different from most of the previous works.

1. To approximate r=φc directly instead ofc. Due to the existence ofφin the time derivative in (2.4), it is not easy to extract the cell averages of c in DG methods (we cannot simply take the test function to be 1). Therefore, we introducer =φc, and approximate r instead ofc.

2. Treat ptas a source in (2.4). We will first solveptin (2.2) and then use the positivity-preserving technique introduced in [47, 48] to construct positive numerical approximations of r.

3. Choose a consistent flux pair for (2.2) and (2.4). In Section 2.2, we will suitably choose the numerical fluxes and prove ¯r <φ, where ¯¯ r and ¯φare the cell averages of r and φ, respectively.

4. The classical bound-preserving limiters (1.4) cannot be applied. Notice that, in the previous point, φ, the upper bound of r, may not be a constant. It is impossible to apply the bound- preserving limiter (1.4) to obtain desired new approximations between 0 andφat the cell inter- faces. For example, take φ(x) = 2−x² on [-1,1] and ¯r = 1.1. We cannot find any θ such that 0≤r(x)˜ ≤φ(x) at x=±1.

5. In this paper, we would like to approximate φby a piecewise linear approximation Φ, such that in cellIj

Φ(x_j±¹

2) =φ(x_j±¹

2),

and we denote this interpolation operator to beI1. Notice that, Φ is a second-order approxima- tion of φ. Since we use second-order DG method, the usage of Φ in the scheme will not kill the accuracy. Moreover, in this paper, we assume φ to be continuous, then Φ is also a continuous function and we denote Φ_j+¹

2 = Φ(x_j+¹

2).

6. Introduce a new limiter. A new limiter will be given to keep the numerical cell average and modify the numerical approximations such that 0≤r≤Φ at the cell interfaces. Sincer and Φ are both linear functions in each cell, then with the new limiter we have 0≤r ≤Φ for all points in the computational domain, which further yieldsc= _Φ^r ∈[0,1].

Now, we are ready to construct the DG scheme. For simplicity, if not otherwise stated, we usep, u, c, r as the numerical approximations from now on. then the DG scheme for (2.2)-(2.4) is the following:

find p, u, r ∈V_h such that for anyξ, η, ζ ∈V_h ( ˜d(r)p_t, ξ) = (u, ξ_x) +

NX−1 j=1

ˆ u[ξ]_j+¹

2 + (q, ξ)_j, (2.5)

(a(c)u, η) = (p, ηx) + XN j=0

ˆ p[η]_j+¹

2, (2.6)

(rt, ζ) = (uc−D(u)cx, ζx) +

NX−1 j=1

c uc[ζ]_j+¹

2 −

NX−1 j=1

{D(u)cx}[ζ]_j+¹

2

−

NX−1 j=1

{D(u)ζx}[c]_j+¹

2 −

NX−1 j=1

˜ α

h[c][ζ]_j+¹

2 + (˜cq−rz₁pt, ζ), (2.7) where c=I1

_r

Φ , ˜d(r) = z₁r+z₂(Φ(x)−r) and (u, v) = R

Ωuvdx. Here we take c as the piecewise linear approximation approximation of _Φ^r such that in each cellIj, we have

c⁺_j

−¹2

= r_j⁺

−¹2

Φ_j−¹

2

, c⁻_j+1 2

= r⁻

j+¹2

Φ_j+¹

2

.

Remark 2.1. In (2.5)-(2.7), the usage of d(r)˜ instead of d(c) is required by the positivity-preserving technique. In the proof of Theorem 2.2, we will subtract (2.5)from (2.7)to obtain a new source similar to the one in (2.7). If we use d(c) in (2.5), then the source in (2.7) should be changed to cΦz₁pt.

In (2.5)-(2.7), ˆu, ˆpandcucare the numerical fluxes. We use alternating fluxes for the diffusion term ˆ

u_j+¹

2 =u⁺

j+¹2

, pˆ_j+¹

2 =p⁻

j+¹2

, j= 1,· · ·, N −1, (2.8) and at the boundary, we choose

ˆ p¹

2 =p⁺1 2

, pˆ_N+¹

2 =p⁻

N+¹₂. For the convection term, we use the following flux

c

uc=u⁺c⁺−α[c], (2.9)

Before we complete this subsection, we would like to introduce the following definition that will be used in the bound-preserving limiter.

Definition 2.1. We say two fluxes cuc,uˆ are consistent ifcuc = ˆu by taking c= 1 in Ω.

The numerical flux pair (cuc,u) given in (2.8) and (2.9) are consistent, and this is required by theˆ bound-preserving technique.

Remark 2.2. There are plenty of flux pairs can be used following the procedures introduced in the next subsection. The proofs are basically the same with some minor changes, so we only list some of them below without more details.

• uˆ=u⁻,pˆ=p⁺, cuc=u⁻c⁻−α[c].

• uˆ= ¹₂(u⁺+u⁻), pˆ= ¹₂(p⁺+p⁻), cuc= ¹₂(u⁺c⁺+u⁻c⁻)−α[c].

2.2 Bound-preserving technique

In this subsection, we consider Euler forward time discretization and apply the bound-preserving technique to construct physically relevant numerical approximations.

For simplicity, we use oj for the numerical approximation o in Ij, and the cell average is ¯oj for o=u, p, c, r. Moreover, we useoⁿto represent the solutionoat time leveln. Takeζ = 1 inI_j in (2.7) forj = 2,· · · , N −1 to obtain the equation satisfied by ¯r,

¯

r_jⁿ⁺¹=H_j^c(r, u, c) +H_j^d(r, u, c) +H_j^s(r,˜c, q, z₁pt), (2.10) where

H_j^c(r, u, c) = 1

3¯rⁿ_j +λ c uc_j−¹

2 −cuc_j+¹

2

, H_j^d(r, u, c) = 1

3¯rⁿ_j −λ

{D(u)cx}j−¹2 + α˜

∆x[c]_j−¹

2 − {D(u)cx}j+¹₂ − α˜

∆x[c]_j+¹

2

, H_j^s(r,˜c, q, z₁pt) = 1

3¯rⁿ_j + ∆t˜cq−rz₁pt,

with λ= _∆x^∆t being the ratio of the time and space mesh sizes, ˜α being another parameter that will be chosen by the positivity-preserving technique, and ˜cq−rz₁pt being the cell average of ˜cq−rz₁pt. We will prove that if ∆tis sufficiently small, thenH_j^c,H_j^d andH_j^s are all positive, and the results are given in the following three lemmas. For simplicity of presentation, if the denominator in a fraction is zero, then the value of the fraction is defined as ∞. Let us consider H_j^c first.

Lemma 2.1. Suppose rⁿ>0 (cⁿ>0), ifα and λsatisfy

α > max

1≤j≤N−1

u⁺

j+¹2

,0

, λ≤ min

2≤j≤N−1







 Φ⁰_m

6α, Φ_j−¹

2

6

α−u⁺_j

−2¹









, (2.11)

where

Φ⁰_m= min

2≤j≤N−1Φ_j+¹

2, then H_j^c(r, u, c)>0.

Proof. It is easy to check H_j^c(r, u, c) = 1

3r¯_jⁿ+λ c uc_j−¹

2 −cuc_j+¹

2

= 1 6

c⁺Φ_j−¹

2 +c⁻Φ_j+¹

2

+λ

u⁺c⁺

j−¹2 −α[c]_j−¹

2 −u⁺c⁺

j+¹₂ +α[c]_j+¹

2

= αλc⁻_j

−¹2

+ 1

6Φ_j−¹

2 +λu⁺_j

−2¹ −αλ

c⁺_j

−¹2

+ 1

6Φ_j+¹

2 −αλ

c⁻

j+¹2

+λ

α−u⁺

j+¹2

c⁺

j+2¹

. Therefore, H_j^c(r, u, c)>0, ifα and λsatisfy condition (2.11).

Now, we proceed to analyzeH_j^d. Following the same analysis in [49] with some minor changes, we can show the following lemma.

Lemma 2.2. Suppose rⁿ>0 (cⁿ>0), then H_j^d(r, u, c)>0 under the condition

˜ α > 1

2D⁰_M, Λ = ∆t

∆x² ≤ min

2≤j≤N−1





Φ_j−¹

2

3D⁻

j+₂¹ + 6˜α, Φ_j+¹

2

6˜α+ 3D⁺

j−¹2



, (2.12)

where

D⁰_M = max

2≤j≤N−1

D⁻

j−¹2

, D⁺

j+¹₂

. Proof. Notice thatc is a linear function in each cell, then

cx− j−¹2

= c⁻

j−¹2 −c⁺

j−³2

∆x , cx+ j−2¹

= c⁻

j+¹₂ −c⁺

j−¹2

∆x =cx−

j+¹₂, cx+ j+₂¹ =

c⁻

j+³₂ −c⁺

j+¹₂

∆x .

Therefore, H_j^d(r, u, s) = 1

3r¯ⁿ_j −λ

{D(u)cx}j−¹2 + α˜

∆x[c]_j−¹

2 − {D(u)cx}j+¹₂ − α˜

∆x[c]_j+¹

2

,

= 1 6Φ_j−¹

2c⁺

j−¹2 −1 2ΛD⁻

j−2¹

c⁻

j−¹2 −c⁺

j−³2

−1 2ΛD⁺

j−¹2

c⁻

j+¹₂ −c⁺

j−¹2

−Λ˜α

c⁺

j−¹2 −c⁻

j−¹2

+1 6Φ_j+¹

2c⁻

j+¹₂ +1 2ΛD⁻

j+¹₂

c⁻

j+¹₂ −c⁺

j−¹2

+1

2ΛD⁺

j+¹₂

c⁻

j+³₂ −c⁺

j+¹₂

+ Λ˜α

c⁺

j+¹₂ −c⁻

j+¹₂

= 1 2ΛD_j⁻

−¹2

c⁺_j

−³2

+ Λ

˜ α−1

2D_j⁻

−¹2

c⁻_j

−¹2

+ 1

6Φ_j−¹

2 +1 2ΛD_j⁺

−2¹ −1 2ΛD⁻

j+¹2 −Λ˜α

c⁺_j

−¹2

+ 1

6Φ_j+¹

2 −1 2ΛD⁺_j

−¹2

+1 2ΛD⁻

j+¹2 −Λ˜α

c⁻

j+¹2

+ Λ

˜ α−1

2D⁺

j+¹2

c⁺

j+¹2

+ 1 2ΛD⁺

j+¹2

c⁻

j+³2

, whereD^±_j

−¹2

=D(u^±_j

−¹2

)≥0.Therefore, we haveH_j^d(r, u, s)>0 if (2.12) is satisfied.

Next, we proceed to studyH_j^s. We use Gaussian quadrature with two points to approximate the cell average of the source. The quadrature points on Ij are denoted as xⁱ_j,i= 1,2. Also, we denote wi as the corresponding weights on the interval [−¹₂,¹₂]. Then we can state the result.

Lemma 2.3. Suppose rⁿ>0 (cⁿ>0), then H_j^s(r,˜c, q, z1pt)>0 under the conditions

∆t≤ 1

6z₁p_M, ∆≤ min

1≤j≤N

min{Φ_j−¹

2,Φ_j+¹

2} 6 maxn

−q(x¹_j),−q(x²_j),0o. (2.13) where

p_M = max

i,j p_t(xⁱ_j),0

. (2.14)

Proof. The cell average of the source is approximated by the Gaussian quadrature, then H_j^s(r, u, c) = 1

3r¯_jⁿ+ ∆t˜cq−rz₁pt= X2 i=1

wi(L¹_i +L²_i), where

L¹_i = 1

6−∆tz1pt(xⁱ_j)

rj(xⁱ_j), L²_i = 1

6rj(xⁱ_j) + ∆t˜cq(xⁱ_j).

Clearly, we have L¹_i ≥0 if (2.13) is satisfied. Moreover,L²_i >0 ifq(xⁱ_j)≥0. We only need to consider the case with q(xⁱ_j)<0. Without loss of generality, we assumeq(x¹_j)<0, then ˜c(x¹_j) =c(x¹_j). Notice that r and care both linear functions in Ij then it is easy to check

r(x¹_j) =µ₁r⁺

j−¹2

+µ₂r⁻

j+₂¹, c(x¹_j) =µ₁c⁺

j−¹2

+µ₂c⁻

j+¹₂, with µ₁= ³⁺₆^√3 and µ₂ = ³⁻₆^√3. Therefore

L²₁ = 1 6(µ1r_j⁺

−¹2

+µ2r_j+⁻ 1 2

) + ∆t(µ1c⁺_j

−¹2

+µ2c⁻_j+1 2

)q(x¹_j)

= 1

6Φ_j−¹

2 + ∆tq(x¹_j)

µ₁c⁺

j−¹2

+ 1

6Φ_j+¹

2 + ∆tq(x¹_j)

µ₂c⁻

j+¹₂. thenL²₁ >0 under (2.13).

Now we study the cells near ∂Ω. For simplicity of presentation, we only demonstrate the results below, and the proof can be obtained exactly the same way in Lemmas 2.1, 2.2 and 2.3 with some minor changes.

Lemma 2.4. Suppose rⁿ₁ >0 (cⁿ₁ >0), then r¯₁ⁿ⁺¹>0 under conditions (2.13) and α > u⁺3

2

, λ≤ Φ³

2

6α, and α >˜ 1 2D⁺3

2

, Λ≤min



 Φ¹

2

3D⁻3 2

,Φ³

2

6˜α



. Similarly, suppose rⁿ_N >0 (cⁿ_N >0), then ¯r_Nⁿ⁺¹ >0 under conditions (2.13) and

λ≤ Φ_N−¹

2

6

α−u⁺_N

−¹2

, and α >˜ 1 2D_N⁻

−¹2

, Λ≤min



 Φ_N−¹

2

6˜α , Φ_N+¹

2

3D⁺_N

−¹2



.

Based on the above lemmas, we can state the following theorem.

Theorem 2.1. Consider the DG scheme (2.5)-(2.7) with Euler forward time discretization. Suppose rⁿ>0 (cⁿ>0), and the parameters α and α˜ are taken to be

α > max

1≤j≤N−1

u⁺

j+¹2

,0

, α >˜ 1 2DM, where

DM = max

1≤j≤N−1

D^±

j+¹₂

. Then r¯_jⁿ⁺¹>0 under (2.13) and

λ≤ min

1≤j≤N−1







 Φm

6α, Φ_j+¹

2

6

α−u⁺

j+¹₂









, Λ≤ min

1≤j≤N





Φ_j−¹

2

6˜α+ 3D⁻

j+¹₂

, Φ_j+¹

2

6˜α+ 3D⁺

j−¹2



,

where

Φ_m= min

1≤j≤N−1Φ_j+¹

2.

Proof. For j = 0 and N, the results was proved in Lemma 2.4. For j = 1,2,· · · , N −1, by (2.10), we only need H_j^c, H_j^d and H_j^s to be positive, which follow directly from Lemmas 2.1, 2.2 and 2.3, respectively.

The above theorem guarantees the positivity of ¯rⁿ⁺¹. However, we still need to show ¯rⁿ⁺¹ ≤Φ,¯ and the result is given below.

Theorem 2.2. Suppose the conditions in Theorem 2.1 are satisfied. Moreover, we assume0≤rⁿ≤Φ and the flux pair (cuc,u)ˆ are consistent, then 0≤r¯ⁿ⁺¹≤Φ,¯ under another condition

∆t≤ 1 6z₂p_M, where p_M was given in (2.14).

Proof. Since the fluxescuc and ˆu are consistent, then cuc+ucc2 = ˆu, wherec2 = 1−c. Take ξ =ζ in (2.5), and subtract (2.7) from (2.5) to obtain

(r_2t, ζ) = (uc₂−D(u)c_2x, ζ_x) +

NX−1 j=1

c uc₂[ζ]_j+¹

2 −

NX−1 j=1

{D(u)c_2x}[ζ]_j+¹

2

−

NX−1 j=1

{D(u)ζx}[c₂]_j+¹

2 −

NX−1 j=1

˜ α

h[c₂][ζ]_j+¹

2 + (˜c₂q−r₂z₂pt, ζ), (2.15) where r₂ = Φ−r and ˜c₂ = 1−˜c. It is easy to check that (2.15) is exactly (2.7) with r, c and z₁ replaced byr₂, c₂ and z₂, respectively. Following the same analyses in Theorem 2.1, we can show that

¯

rⁿ⁺¹₂ >0 under the conditions given in this theorem, which further implies ¯rⁿ⁺¹ <Φ.¯

With Theorems 2.2, the numerical cell average ¯rⁿ we constructed is between 0 and ¯Φ. However, the numerical approximation rⁿ may be negative or larger than Φ. Therefore, we also need to apply some limiter to rⁿ and the procedure is given below. For simplicity, we drop the superscriptn in all the numerical approximations.

1. Set up a small numberǫ= 10⁻¹³.

2. If ¯r≤ǫ, taker = ¯r. If ¯r₂ ≤ǫ, taker = Φ−r¯₂. Then skip the following steps.

3. Definemj = min{r_j⁺

−¹2

, r⁻

j+¹2}. Ifmj ≥0 then proceed to the next step. Otherwise, without loss of generality, we assumer⁺_j

−¹2

<0, then define

˜ r⁺_j

−¹2

=ǫ, r˜⁻

j+¹2

=r⁻

j+¹2 −ǫ+r⁺_j

−¹2

and use ˜r⁺_j

−¹2

and ˜r⁻

j+2¹

to construct a new approximation, also denoted asr.

4. Apply the previous step forr₂= Φ−r and construct a new approximation r.

Remark 2.3. The above algorithm keeps the second-order accuracy. In step 3, we assume r⁺_j

−¹2

<0 and 0≤Φ_j−¹

2 −r⁺_j

−2¹ ≤Ch², then it is very easy to check0≤Φ_j−¹

2 −˜r⁺_j

−2¹ ≤Ch².

Remark 2.4. For an n-component fluid (n≥3), we denote the concentration of the i’th component to be ci, and letri =φci.Follow the same analysis in Theorem 2.2, we can show r¯i≥0, i= 1,2,· · · , n and Pn

i=1r¯i = ¯Φ. However, the limiter is not easy to construct. We will work on this problem in the future.

Remark 2.5. The extension to high-order schemes is not straightforward as demonstrated in [49].

Recently, Chen [4] introduced the third-order maximum-principle preserving direct DG methods for convection-diffusion equations which can be applied to this problem. Another approach is to applied the flux limiter [34, 33]. However, the accuracy was demonstrated by numerical experiments only.

2.3 High order time discretizations

All the previous analyses are based on first-order Euler forward time discretization. We can also use strong stability preserving (SSP) high-order time discretizations to solve the ODE system w_t=Lw.

More details of these time discretizations can be found in [25, 24, 14]. In this paper, we use the third-order SSP Runge-Kutta method [25]

w⁽¹⁾ = wⁿ+ ∆tL(wⁿ), w⁽²⁾ = 3

4wⁿ+1 4

w⁽¹⁾+ ∆tL(w⁽¹⁾)

, (2.16)

wⁿ⁺¹ = 1

3wⁿ+2 3

w⁽²⁾+ ∆tL(w⁽²⁾) , and the third order SSP multi-step method [24]

wⁿ⁺¹= 16

27(wⁿ+ 3∆tL(wⁿ)) +11 27

wⁿ⁻³+12

11∆tL(wⁿ⁻³)

. (2.17)

Since an SSP time discretization is a convex combination of Euler forward and the algorithm is to construct positive r and Φ−r, hence by using the limiter designed in section 2.2, the numerical solution obtained from the full scheme is also physically relevant.

3 Numerical algorithm in two space dimensions

In this section, we proceed to construct the bound-preserving DG scheme for the compressible miscible displacement in porous media in two space dimensions.

3.1 The DG scheme

We first explain the notations that will be used in this section. Denote Ω = [0,2π]×[0,2π] to be the computational domain. In this paper, we consider rectangular meshes only, and the techniques for triangular meshes will be discussed in the future. Let 0 = x¹

2 < · · · < x_N

x+¹₂ = 2π and 0 = y¹

2 <

· · ·< y_N

y+¹₂ = 2π be the grid points in x andy directions, respectively. DefineIi = (x_i−¹

2, x_i+¹

2) and J_j = (y_j−¹

2, y_j+¹

2). Let K_ij =I_i×J_j, i= 1,· · · , N_x, j = 1,· · · , N_y,be a partition of Ω and denote Ωh ={Kij}.For simplicity, if not otherwise stated, we always useK to denote the cell. We use Γ for all the cell interfaces, and Γ₀ = Γ\∂Ω for all the interior ones. For any e∈ Γ, denote |e| to be the length of e. Moreover, we define ne=nx = (1,0) if eis parallel to the y-axis while ne=ny = (0,1) if eis parallel to the x-axis. Further more, we denote∂Ω₊={e∈∂Ω :n_e =n}, where n is the unit outer normal of ∂Ω, and ∂Ω₋ = ∂Ω\∂Ω₊. The mesh sizes in the x and y directions are given as

∆x_i =x_i+¹

2 −x_i−¹

2 and ∆y_j =y_j+¹

2 −y_j−¹

2, respectively. For simplicity, we assume uniform meshes and denote ∆x= ∆xi and ∆y= ∆yj.However, this assumption is not essential. Following [49], we use second-order DG scheme. For rectangular meshes, it is impossible to construct a continuous piecewise linear interpolation of φby using P¹ polynomials. Therefore, the finite element space is chosen as

Wh ={z:z|^K ∈Q¹(K),∀K ∈Ωh},

where Q¹(K) denotes the space of tensor product of linear polynomials in K. Given v ∈ Wh, we denote v⁺

i−¹2,j,v⁻

i+¹₂,j,v⁺

i,j−¹2

,v⁻

i,j+₂¹ to be the traces ofvon the four edges of K_ij, respectively, and use [v] =v⁺−v⁻ and {v}= ¹₂(v⁺+v⁻) as the jump and average of v at the cell interfaces, respectively.

For simplicity, for any e∈∂Ω₋, we define v⁻|^e= 0. Similarly, for any e∈∂Ω+, we define v⁺|^e= 0.

Moreover, we denote Φ∈W_h to be an interpolation of φsuch that in each cell K, Φ =φat the four corners, and we denote this interpolation operator to be I1. It is easy to see that Φ is a globally continuous function on Ω and, for simplicity, define Φ_i+¹

2,j+¹2 = Φ(x_i+¹

2, y_j+¹

2) =φ(x_i+¹

2, y_j+¹

2) for all i= 0,· · · , Nx, j= 0,· · ·, Ny.

We also usep, c,u as the numerical approximations, then the DG scheme is to findp, r∈W_h and

u∈W_h =Wh×Wh such that for anyξ, ζ ∈Wh andη∈W_h, ( ˜d(r)p_t, ξ) = (u,∇ξ) + X

e∈Γ⁰

Z

e

ˆ

u[ξ]·n_eds+ (q, ξ), (3.1)

(a(c)u,η) = (p,∇ ·η) +X

e∈Γ

Z

e

ˆ

p[η·ne]ds, (3.2)

(r_t, ζ) = (uc−D(u)∇c,∇ζ) + (˜cq−rz₁p_t, ζ) + X

e∈Γ⁰

Z

eucc·n_e[ζ]ds

−X

e∈Γ⁰

Z

e

{D(u)∇c·n_e}[ζ] +{D(u)∇ζ·n_e}[c] + α˜

|e|[c][ζ]

ds, (3.3)

where

c=I1

nr Φ

o, d(r) =˜ z₁r+z₂(Φ−r), (u, v) = Z

Ω

uvdxdy.

In (3.1)-(3.3), ˆp, uˆ and cuc are the numerical fluxes. We also use alternating fluxes for the diffusion terms, and for any e∈Γ₀

ˆ

u|^e=u⁺|^e, pˆ|^e=p⁻|^e, and on ∂Ω, we take

ˆ

p|^e =p⁻|^e, ∀ e∈∂Ω+, pˆ|^e=p⁺|^e, ∀ e∈∂Ω₋. For the convection term, we use

c

uc=u⁺c⁺−α[c], (3.4)

which is consistent with ˆu. Here α and ˜α are two positive constants to be chosen by the bound- preserving technique. Similar to Remark 2.2, there are some other choices of the numerical fluxes, yet we will not discuss them in detail.

3.2 Bound-preserving technique

In this section, we apply the bound-preserving technique to the DG scheme, and construct physically relevant numerical approximations. We consider Euler-forward time discretization only, and the high- order ones were discussed in Section 2.3. For simplicity of presentation, we only discuss the techniques for cells away from∂Ω, while the boundary cells can be analyzed following the same lines. We useoij

for the numerical approximation oinKij and the cell average is ¯oij. Moreover, we useoⁿto represent the solution oat time leveln.

In (3.3), we takeζ = 1 in Kij to obtain the equation satisfied by the cell average of r,

¯

r_ijⁿ⁺¹ =H^c(r,u, c) +H_x^d(r,u, c) +H_y^d(r,u, c) +H^s(r,˜c, q, z₁p_t), (3.5)

where

H^c(r,u, c) = 1 3¯rⁿ_ij+λ

Z

Jj

c u₁c_i−¹

2,j−uc₁c_i+¹

2,jdy+ Z

Ii

c u₂c_i,j−¹

2 −uc₂c_i,j+¹

2dx

! , H_x^d(r,u, c) = 1

6¯rⁿ_ij−λ Z

J_j{D₁₁cx+D₁₂cy}i−¹2,j+ α˜

|Jj|[c]_i−¹

2,j

− {D₁₁cx+D₁₂cy}_i+¹₂,j− α˜

|J_j|[c]_i+¹

2,jdy

, H_y^d(r,u, c) = 1

6¯rⁿ_ij−λ Z

I_i{D₂₁cx+D₂₂cy}i,j−¹2 + α˜

|Ii|[c]_i,j−¹

2

− {D₂₁cx+D₂₂cy}_i,j+¹

2 − α˜

|I_i|[c]_i,j+¹

2dy

, H^s(r,˜c, q, z₁p_t) = 1

3¯rⁿ_ij+ ∆t˜cq−rz₁p_t,

with λ = _∆x∆y^∆t and u = (u₁, u₂)^T. To continue, we use 2-point Gaussian quadratures to approx- imate the integrals given above. The Gaussian quadrature points on I_i and J_j are denoted by x¹_i, x²_i and n

y_j¹, y_j²o

, respectively. The corresponding weights on the interval [−¹₂,¹₂] are de- noted as w1 and w2. Denote λ1 = _∆x^∆t and λ2 = _∆y^∆t, then

H^c(r,u, c) = 1

3r¯ⁿ_ij+λ₁ X2 β=1

w_βh c u₁c_i−¹

2,j,β−uc₁c_i+¹

2,j,β

i +λ₂ X2 β=1

w_βh c u₂c_i,j−¹

2,β−uc₂c_i,j+¹

2,β

i,

whereuc₁c_i−¹

2,j,β=uc₁c_i−¹

2,j(y_j^β) is a point value in the Gaussian quadrature on the edgee={x_i−¹

2}×Jj. Likewise for the other point values. As the general treatment, we rewrite the cell average on the right hand side as

¯ rⁿ_ij =

X2 β=1

wβ

2

r⁺

i−¹2,j,β+r⁻

i+¹₂,j,β

= X2 β=1

wβ

2

r⁺

i,j−2¹,β+r⁻

i,j+₂¹,β

. Based on the above,

H^c(r,u, c) = X2 β=1

w_β

λ₁ 6(λ1+λ2)

r⁺

i−¹2,j,β+r⁻

i+¹2,j,β

+λ₁

c u₁c_i−¹

2,j,β−uc₁c_i+¹

2,j,β

+ X2 β=1

w_β

λ₂ 6(λ₁+λ₂)

r⁺

i,j−¹2,β+r⁻

i,j+¹₂,β

+λ₂

c u₂c_i,j−¹

2,β−uc₂c_i,j+¹

2,β

.

Following the same proof in Lemma 2.1 with some minor changes, we have Lemma 3.1. Suppose rⁿ>0 (cⁿ>0) and the parameter α is taken to be

α > max 1≤i≤Nx−1, 1≤j≤N_y−1,

β= 1,2

{u₁⁺

i+¹₂,j,β, u₂⁺

i,j+¹₂,β,0}, (3.6)