High-Order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes

High-Order bound-preserving discontinuous Galerkin methods for wormhole propagation on triangular meshes ^∗

Ziyao Xu

^†

, Yang Yang

^‡

, Hui Guo

^Ÿ

Abstract

Wormhole propagation, arising in petroleum engineering, is used to describe the distribution of acid and the increase of porosity in carbonate reservoir under dissolution of injected acid. The important physical features of porosity and acid concentration include their boundedness between 0 and 1, as well as the monotone increasing for porosity. How to keep these properties in the simulation is crucial to the robustness of the numerical algorithm. In this paper, we propose high-order bound-preserving discontinuous Galerkin methods to keep these important physical properties. The main technique is to introduce a new variable r to replace the original acid concentration and use a consistent ux pair to deduce a ghost equation such that the positive-preserving technique can be applied on both original and deduced equations. A high-order slope limiter is used to keep a polynomial upper bound which changes over time forr. Moreover, the high-order accuracy is attained by the ux limiter. Numerical examples are given to demonstrate the high-order accuracy and bound-preserving property of the numerical technique.

Key Words: wormhole propagation, bound-preserving, high-order, discontinuous Galerkin method, triangular meshes, ux limiter

1 Introduction

As an important technique of enhanced oil recovery (EOR), acid treatment has been widely practiced in carbonate reservoir to improve the productivity of oil wells. In this technique, acid is injected into wells to dissolve the nes deposed in wellbore and the rock near the wellbore. By doing so, the permeability and porosity of the rock close to a well can be increased prominently, which facilitates oil ow into production well and thereby improves the production rate of oil.

∗The rst and second authors were supported by the NSF grant DMS-1818467 and the last author was supported by National Natural Science Foundation of China Grants 11571367 and the Fundamental Research Funds for the Central Universities 18CX02021A.

†Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931. E-mail: [email protected]

‡Department of Mathematical Sciences, Michigan Technological University, Houghton, MI 49931. E-mail: [email protected]

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

However, the eciency of this technique has a strong relevance with the dissolution patterns which depend on the injection rate. With a very low injection rate, the acid only dissolves the face of wellbore since it will be consummated all before they get into deeper region and this scenario is called face dissolution pattern.

In contrast, with a very high injection rate, the acid can be pushed uniformly into the wellbore region with certain depth and this results is the so-called uniform dissolution pattern. In addition to the above two extreme cases, with an appropriate injection rate, wormhole patterns can be formed as the injected acid in the rock tends to ow through the paths with high permeability and porosity, which causes the permeability and porosity of these path to be further increased under the dissolution of acid, and facilitate more acid to ow through. Therefore, under optimal injection rate, maximum number of narrow channels with high conductivity will be formed in the rocks after the acidizing process. These highly conducting channels, also known as wormholes because of its shape, can build a good connectivity between reservoir and wellbore, and improve the productivity of oil well enormously. Because of the important role that wormhole plays in improving productivity, a lot of research works have been taken to investigate the formation and propagation of wormholes.

In the early days, researchers investigated the wormhole propagation phenomenon by means of exper- iments [11, 5]. Later, several mathematical models, such as dimensionless model, capillary tube model, network model, and continuum models, were established to help people understand the process of wormhole propagation. Among these models, the most popular one is the two-scale continuum model developed by Panga et al. in [16], where the authors proposed a partial dierential equations (PDE) system to describe the formation and propagation of wormholes. There were a lot of follow-up works based on this model. In [33], the authors analyzed the front instability of wormhole propagation theoretically and numerically. In [15], Maheshwari et al. presented a 3D simulation for this model. A parallel simulation was conducted by Wu et al. in [21] under a modication of ow equation. In [1], the authors studied the numerical-simulation approach for a modied model. Later, Wei et al. extended this model from single phase to two-phase ow in [19] and discussed the simulation results. Besides the above, many researchers designed specic numerical schemes for this kind of models as well. In [12], the authors constructed a conservative scheme for ow and transport based on mixed nite element method. Later, Li et al. applied nite dierence methods to this problem in [13, 14]. Recently, the discontinuous Galerkin (DG) method was applied to this model in [8].

In all the above works, some theoretical works, such as the stability and error estimates, were established under dierent norms. However, to the best of our knowledge, no works has been undertaken to preserve the boundedness of porosity and concentration of acid without loss of mass conservative. In fact, the bound- edness of these variables is essential for the stability of numerical simulations. Firstly, the rate of change of porosity φ in this model depends on the concentration of acid c_f. If the exact solutions contain large gradients or even discontinuities, the numerical approximations of cf can be negative, which further leads to φ < 0 in some regions with low porosity. Secondly, the oscillations of φ itself near the wormhole may also result in negative values. Both of the above two cases will bring a negative coecient in the diusion term of the transport equation, leading to ill-posedness of the problems, and nally cause the blow-up of the numerical simulations. We will demonstrate this feasibility by a numerical example in Section 6 and show how bound-preserving technique can prevent the blow-up phenomenon. Moreover, as we will see in the later section, many coecients in the model appear as functions ofφ, which requireφto take values in the physically relevant range[0,1]as well. To construct high-order bound-preserving technique, we have to apply suitable limiters to modify the numerical approximations. Therefore, we would like to use DG methods.

The DG methods become increasingly popular due to their good stability, high-order accuracy, and ex-

ibility on h-p adaptivity. In 2010, the genuinely maximum-princip le-satisfying high-order DG schemes were constructed for conservation laws on rectangular meshes in [29] by Zhang and Shu. The basic idea is to take the test function to be1in each cell to yield an equation satised by the cell average of the target variabler, and prove the desired boundedness of the cell average¯r. Then a slope limiter which do not aect accuracy and mass conservation can be used to modify the variablerto obtain a new oner˜= ¯r+θ(r−¯r)such that˜rhas the physically relevant bounds. In the case that the variableronly need a lower bound zero, this technique is also called positivity-preserving technique. The physically positivity-preserving and bound-preserving numerical schemes have been actively studied since then. In 2012, this technique has been successfully ex- tended to triangular meshes in [30], where the general criteria for quadrature rule on triangular elements was proposed. After that, this technique was applied to many problems including compressible Euler equations with source terms [31], hyperbolic equations involving δ-singularities [26, 27], relativistic hydrodynamics [17], extended MHD equations [34], shallow water equations [22], etc. For convection-diusion equations, the genuinely second-order maximum-principle-preserving technique were introduced in [32]. Subsequently, the extension to third-order or even higher order bound-preserving techniques for parabolic equations were also developed in [25, 2, 4]. Besides the above, the ux limiter [24, 23, 10] can also be used to obtain the high-order accuracy and maintain the boundedness. However, with the ux limiters we have to modify the numerical uxes, hence the accuracy is not easy to analyze. Recently, in [9, 3], the authors studied miscible displacements in porous media and applied the techniques introduced in [32, 10, 23, 24] to preserve the two bounds, 0 and 1, of the volumetric fractions. In this paper, we will construct high-order bound-preserving DG schemes for the porosity of the rocksφand the concentration of the acidc_f. However, there are signif- icant dierences from most of the previous techniques. First of all, most of the problems in [24, 29] satisfy maximum-principles while the concentration of acid c_f does not. To solve this problem, we derive a ghost equation satised byc= 1−cf and apply the positivity-preserving technique to bothcf andc. Secondly, the high-order positivity-preserving technique in this paper is based on the ux limiter [23, 10]. The basic idea is to combine higher order and lower order uxes to construct a new one which can yield positive numerical cell averages. However, for triangular meshes, rst-order uxes are not easy to construct. Therefore, we will consider the second-order ux as the lower order one. Moreover, to obtain the equation satised by the cell averages, we need to numerically approximate r =φcf instead of cf. By doing so, the upper bound of r is not a constant and the traditional slope limiter may fail to work [9]. Therefore, a new bound-preserving limiter will be introduced. A similar obstacle appeared in the design of high-order bound-preserving methods for general relativistic hydrodynamics, see [20] for the discussion. Finally, dierent from [9, 3], the porosity is increasing and less than 1. Therefore, the upper bound ofris changing during time evolution and special techniques will be introduced to makeφto be physically relevant. In summary, the whole algorithm can be separated into four parts. We rst apply positivity-preserving technique to obtain positiveφtand use which as another source to nd the velocity and pressure. Then apply the positivity-preserving technique again to φ and cf simultaneously to obtain positive numerical cell averages by the ux limiter [23, 10]. Subse- quently, we choose consistent ux pair [9, 3] with suitable parameters in the ux limiter in the concentration and pressure equations to obtain the positivity of1−cf. Finally, we introduce suitable limiters to obtain physically relevant numerical approximations.

The rest of the paper is organized as follows. In Section 2, we introduce the mathematical model of wormhole propagation. In Section 3, we establish the DG scheme used in this paper. In Sections 4 and 5, we construct the second-order bound-preserving scheme and then extend it to high-order spatial and time discretizations. Some numerical examples are given in Section 6. We will end in Section 7 with some

concluding remarks.

2 Mathematical model

Let the computational domainΩ = [0,2π]×[0,2π]and simulation timeJ = [0, T], the mathematical model of the wormhole propagation is given as follows:

∂φ

∂t +∇ ·u=f, (x, y)∈Ω, 0< t≤T, (2.1)

u=−κ(φ)

µ ∇p, (x, y)∈Ω, 0< t≤T, (2.2)

∂(φcf)

∂t +∇ ·(ucf) =∇ ·(φD∇cf) +kcav(cs−cf) +fIcI−fPcf, (2.3)

∂φ

∂t = αk_ca_v(c_f−c_s) ρs

, (x, y)∈Ω, 0< t≤T, (2.4)

whereφis the porosity which is dened as the percentage of the empty space in a rock,κis the permeability that measures the ability for a rock to allow uid to pass through it,uis the Darcy's velocity dened as the volume of ow crossing a unit across-section per unit time, pis the pressure of uid in porous media, and µis the viscosity of uid. f =fI−fP is the external volumetric ow rate with fI = max{f,0}being the injection ow rate andf_P =−min{f,0}being the production ow rate. c_f,c_sandc_I are the concentrations of acid in the uid phase, the uid-solid interface and in the injected ow, respectively. Dis the dispersion tensor for the acid in porous media and k_c is the local mass-transfer coecient. a_v is the interfacial area available for reaction, ρs is the density of the rock and αis the dissolving constant of the acid, dened as grams of solid dissolved per mole of acid reacted. Moreover, in the case of rst order kinetic reaction, the concentrationcsof acid in the uid-solid interface have a simple relationship with cf:

c_s= c_f 1 +ks/kc

,

wherek_sis the kinetic constant for reaction. The coecientsκanda_v are functions ofφdened as κ

κ0

= φ φ0

φ(1−φ₀) φ0(1−φ)

² , a_v

a0

= 1−φ 1−φ0

, (2.5)

respectively, where κ₀, a₀, and φ₀ are the initial values for κ, a_v, φ. Throughout this paper, the values µ, kc, ks, α, ρsare positive constants, D, f, fI, fP, cI are known functions independent of time andφ,u, p, cf, are unknown time-dependent variables.

We consider impermeable boundary conditions

u·n= 0, (D∇c−cu)·n= 0,

where n is the unit outer normal of the boundary∂Ω. The initial concentration and porosity are given as c_f(x, y,0) =c₀(x, y), φ(x, y,0) =φ₀(x, y), (x, y)∈Ω.

Before we nish this section ,we would like to make an important reasonable hypothesis for the initial porosity: 0< φ_?≤φ₀(x, y)≤φ^?<1.

3 The DG scheme

In this section, we will construct the DG scheme for wormhole propagation on triangular meshes. We rst demonstrate the notations to be used throughout the paper.

Consider a regular triangulation Ωhof domain Ω, i.e. ∃ρ >0, such that diam(BK)≥ρdiam(K),∀K∈ Ω_h, whereB_Kis the largest ball contained inK. For any triangleK∈Ω_h, we denote the three edges ofKto beeⁱ_K (i= 1,2,3), with corresponding lengths`ⁱ_K (i= 1,2,3), unit outer normal vectorsν_Kⁱ (i= 1,2,3)and neighboring elements K_i(i = 1,2,3). We denote Γ =S

K∈Ω_h{e|e∈∂K} to be the set of all cell interfaces andΓ0= Γ\∂Ωhas all the interior ones. Set a predetermined constant unit vectorν0 which is not parallel to any edge eand dene ne as the unit normal vector of each edge e∈Γ such thatne·ν0 >0. For any discontinuous function v (scalar or vector) crossing edgee, let v^±_e denote its traces on eevaluated fromK orKi . The⁰±⁰ for each edgeeⁱ_K in the cellK is determined by the inner product ofν_Kⁱ andν0as follows:

v_e⁻=vK, v_e⁺=vK_i, ifν0·ν_Kⁱ >0, v_e⁺=v_K, v_e⁻=v_Ki, ifν₀·ν_Kⁱ <0.

Moreover, we dene the jump and average ofv (either a scalar or a vector) on the cell interfaceeas [v]_e=v⁺_e −v⁻_e, {v}_e=1

2(v_e⁺+v_e⁻).

The nite element spaces are chosen as

Vh={v:v|K∈P^k(K), ∀K∈Ωh}andWh=Vh×Vh,

whereP^k(K)denotes the space of polynomials of degree at mostkinK. Then the semidiscrete DG scheme for (2.1) - (2.4) can be written as: ndφ, r, p∈Vh andu∈Wh such that for anyζ, ξ, v ∈Vh andη∈Wh, the following equations hold:

∂φ

∂t, ζ

= (u,∇ζ) + X

e∈Γ₀

Z

e

uˆ·ne[ζ]ds+ (f, ζ), (3.1)

(a(φ)u,η) = (p,∇ ·η) +X

e∈Γ

Z

e

ˆ

pn_e·[η]ds, (3.2)

∂r

∂t, ξ

= (ucf−φD∇cf,∇ξ) + X

e∈Γ₀

Z

eudcf·ne[ξ]ds+ (fIcI−fPcf −B1(φ)cf, ξ)

−X

e∈Γ0

Z

e

{φD(u)∇cf} ·n_e[ξ] +{φD(u)∇ξ} ·n_e[c_f] + α˜

|e|[c_f][ξ]

ds, (3.3) ∂φ

∂t, v

= (B2(φ)cf, v), (3.4)

where

a(φ) = µ

k, B₁(φ) = a0(1−φ)kskc

(1−φ0)(ks+kc), B₂(φ) = αa0(1−φ)kskc

ρs(1−φ0)(ks+kc).

Moreover, we use a new variable r instead ofφcf on the left hand side of (3.3), and dene cf as the L²- projection of _φ^r ifk≥2, while take c_f to be the interpolation of ^r_φ at the three vertices in each triangleK ifk= 1.

Following the idea in [9, 3], we take a consistent ux pairu,ˆ ucdf in the sense thatuˆ =ucdf whencf = 1. The consistent ux pair is used in the construction of the bound-preserving techniques. The numerical uxes u,ˆ ucdf andpˆin (3.1)-(3.4) are chosen as

u|ˆ e={u}e, p|ˆe={p}e, ucdf|e={ucf}e−α[cf]ene, ife∈Γ0,

u|ˆ e= 0, p|ˆe=pK, ucdf|e= 0, ife∈∂Ω∩∂K. (3.5) In the DG schemes, we introduced two penalty parametersαandα˜. These two parameters will be chosen by the bound-preserving technique.

4 Second-order bound-preserving schemes

In this section, we will construct second-order bound-preserving scheme with forward Euler time discretiza- tion. High-order time discretizations will be discussed in the next section. At time level n, we assume φ⁰< φⁿ<1 and0≤rⁿ≤φⁿ, and would like to construct physically relevant numerical approximations at time leveln+ 1, i.e. φⁿ≤φⁿ⁺¹<1and0≤rⁿ⁺¹≤φⁿ⁺¹.

At time leveln, we will rst solve (3.4) forφⁿ_t, then substitute which to the left-hand side of (3.1). With forward Euler time discretization, (3.1), (3.3) and (3.4) can be written as

φⁿ⁺¹−φⁿ

∆t , ζ

= (u,∇ζ) + X

e∈Γ0

Z

e

uˆ·ne[ζ]ds+ (f, ζ), (4.1)

rⁿ⁺¹−rⁿ

∆t , ξ

= (ucf−φD∇cf,∇ξ) + X

e∈Γ0

Z

e

udcf·ne[ξ]ds+ (fIcI−fPcf−B1(φ)cf, ξ)

−X

e∈Γ0

Z

e

{φD(u)∇cf} ·n_e[ξ] +{φD(u)∇ξ} ·n_e[c_f] + α˜

|e|[c_f][ξ]

ds, (4.2) φⁿ⁺¹−φⁿ

∆t , v

= (B₂(φ)c_f, v), (4.3)

with all the superscriptnon the right hand sides being omitted for simplicity.

Remark 4.1. After the time discretization, there are two discrete evolution equations forφ. We use (4.3) to nd φⁿ⁺¹, then the discrete evolution for φin (4.1) is treated as another source.

Because of the usage of consistent ux pairuˆ anducd_f, we can get a ghost equation forr₂ by subtracting (4.2) from (4.1) and introducing ghost variablesc2= 1−cf, c2I = 1−cI, r2=φc2,

r₂ⁿ⁺¹−rⁿ₂

∆t , ξ

= (uc2−φD∇c2,∇ξ) +X

e∈Γ0

Z

eudc2·ne[ξ]ds+ (fIc2I−fPc2+B1(φ)cf, ξ)

−X

e∈Γ₀

Z

e

{φD(u)∇c2} ·ne[ξ] +{φD(u)∇ξ} ·ne[c2] + α˜

|e|[c2][ξ]

ds. (4.4) Therefore, though we solve (4.1) and (4.2) in the real computation, we analyze (4.2) and (4.4) instead of the former pair as the two forms are equivalent.

The second-order bound-preserving scheme is built and analyzed based on (4.2), (4.4) and (4.3).

In this paper, we use the quadrature rule of orderk proposed in [28] to compute the integral in cells, and

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•

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Figure 1: Distribution of quadrature points fork= 1 andk= 2.

use the correspondingk+ 1 points Gaussian quadrature rule to evaluate integration on cell interfaces. The quadrature rule of orderkhas the following crucial properties:

• All of the quadrature points lie in the cells with positive weights,

• The quadrature points containsk+ 1Gaussian quadrature points in each of its edges,

• It is exact for polynomials up to degree2k−1,

The distribution of quadrature points in the case of k = 1 and k = 2 are shown in Figure 1. We denote x_i,β, β = 1,2,· · · , k+ 1, as the quadrature points on eⁱ_K with w˜_β being the corresponding weights, and denote xγ, γ = 1,2,· · · , L, as the quadrature points in cell K with ωˆγ being the corresponding weights.

Moreover, We denote ω_β, β = 1,2,· · ·, k+ 1, as the k+ 1 Gaussian quadrature weights on the reference interval[−¹₂,¹₂]. Based on the above notations, we dene the values ofo(o=r, c, φ, p,· · ·)at the quadrature points aso^i,β_K =o(x_i,β)along the boundary ofK ando^γ_K =o(x_γ)in cellK.

In (3.4), we takev= 1in K to obtain the equation satised by the cell average ofφ:

φ¯ⁿ⁺¹_K = ¯φⁿ_K+4tB₂(φⁿ)c_f. (4.5) We will demonstrate how to preserve the monotonicity and the upper bound ofφ¯ⁿ⁺¹_K in the following lemma:

Lemma 4.1. Given 0 ≤rⁿ ≤ φⁿ (0 ≤cⁿ_f ≤ 1) and φⁿ < 1, we have φ¯ⁿ_K ≤ φ¯ⁿ⁺¹_K < 1, if the time step satises

4t < B₃₀⁻¹, (4.6)

whereB30 is a constant dened as

B30= αa0kskc

ρs(1−φ^?)(ks+kc). Proof. Dene B₃(x) = _ρ ^αa0kskc

s(1−φ0(x))(ks+kc). ThenB₃(x) is independent of timet and it's easy to check that B2(x, φ) =B3(x)·(1−φ)≤B30·(1−φ).

Applying quadrature rule in [28] to (4.5) with enough algebraic orderk, we have φ¯ⁿ⁺¹_K = ¯φⁿ_K+4tB2(φⁿ)cf

= ¯φⁿ_K+4t





3

X

i=1 k+1

X

β=1

ˆ

wβ(B3)^i,β_K (1−φ^i,β_K )(cf)^i,β_K +

L

X

γ=1

ˆ

wγ(B3)^γ_K(1−φ^γ_K)(cf)^γ_K





≥φ¯ⁿ_K

under the assumption0≤cⁿ_f ≤1and φⁿ_K <1. Moreover, we have

φ¯ⁿ⁺¹_K = ¯φⁿ_K+4t





3

X

i=1 k+1

X

β=1

ˆ

wβ(B3)^i,β_K (1−φ^i,β_K )(cf)^i,β_K +

L

X

γ=1

ˆ

wγ(B3)^γ_K(1−φ^γ_K)(cf)^γ_K





≤φ¯ⁿ_K+4t





3

X

i=1 k+1

X

β=1

ˆ

w_β(B₃)^i,β_K (1−φ^i,β_K ) +

L

X

γ=1

ˆ

w_γ(B₃)^γ_K(1−φ^γ_K)





≤φ¯ⁿ_K+4t





3

X

i=1 k+1

X

β=1

ˆ

w_βB₃₀(1−φ^i,β_K ) +

L

X

γ=1

ˆ

w_γB₃₀(1−φ^γ_K)





= ¯φⁿ_K+4tB30





3

X

i=1 k+1

X

β=1

ˆ

wβ(1−φ^i,β_K ) +

L

X

γ=1

ˆ

wγ(1−φ^γ_K)





= ¯φⁿ_K+4tB30(1−φ¯ⁿ_K).

Thusφ¯ⁿ⁺¹_K <1under the condition (4.6).

The bound-preserving property for ¯rⁿ⁺¹_K is relatively dicult to derive. Therefore, instead of obtaining 0≤r¯_Kⁿ⁺¹≤φ¯ⁿ⁺¹_K directly, we apply the positivity-preserving technique tor¯ⁿ⁺¹_K and ¯r_2Kⁿ⁺¹ in (4.2) and (4.4), respectively, which further yields0≤r¯_Kⁿ⁺¹ ≤φ¯ⁿ⁺¹_K due to the fact that r¯ⁿ⁺¹_K + ¯r_2Kⁿ⁺¹ = ¯φⁿ⁺¹_K . To construct the positivity-preserving technique, in (4.2), we takeξ= 1 inK to obtain the equation satised by the cell average ofr

¯

r_Kⁿ⁺¹=H_K^c(r, c_f,u) +H_K^d(r, c_f,u, φ) +H_K^s(r, c_f, c_I, f_P, f_I, φ), (4.7) where

H_K^c(r, cf,u) = 1 3r¯ⁿ_K−λ

3

X

i=1

Z

eⁱ_Kucdf·ν_Kⁱ ds, (4.8)

H_K^d(r, cf,u, φ) = 1 3r¯ⁿ_K+λ

3

X

i=1

Z

eⁱ_K

{D(u)∇c} ·ν_Kⁱ + α˜

`ⁱ_K[c]ne·ν_Kⁱ

ds, (4.9)

H_K^s(r, cf, cI, fP, fI, φ) = 1

3r¯ⁿ_K+4tfIcI−fPcf−B1(φ)cf, (4.10) withλ=_|K|^4t being the ratio of time step and area of triangular elementK, andfPcf−fIcI−B1(φ)cf being the cell average off_Pc_f−f_Ic_I−B₁(φ)c_f onK. We will demonstrate the positivity-preserving property for

¯

rⁿ⁺¹_K by collecting the following three lemmas.

Lemma 4.2. Given rⁿ>0 (cⁿ_f >0), we have H_K^s(r, cf, cI, fP, fI, φ)>0, if the time step satises 4t≤ φ?

6f_{P M}, 4t≤ φ?

6B₁(φ_?), (4.11)

where

f_{P M} = max

i,β,γ{(fP)^i,β_K ,(fP)^γ_K}.

Proof. We can split (4.10) as H_K^s =4tfIc_I+

1

6r¯_Kⁿ − 4tfPc_f

+ 1

6¯r_Kⁿ − 4tB1(φ)c_f

:=L₁+L₂+L₃.

It is easy to check thatL1 =4tfIcI ≥0. We only need to consider L2 and L3. Applying quadrature rule in [28] with enough algebraic orderk toL₂ andL₃, respectively, we can get

L2= 1

6r¯ⁿ_K− 4tfPcf

= 1 6





3

X

i=1 k+1

X

β=1

˜ w_βr_K^i,β+

L

X

γ=1

ˆ w_γr_K^γ



− 4t





3

X

i=1 k+1

X

β=1

˜

w_β(f_P)^i,β_K (c_f)^i,β_K +

L

X

γ=1

ˆ

w_γ(f_P)^γ_K(c_f)^γ_K





=

3

X

i=1 k+1

X

β=1

˜ wβ

1

6r_K^i,β− 4t(fP)^i,β_K (cf)^i,β_K

+

L

X

γ=1

ˆ wγ

1

6r_K^γ − 4t(fP)^γ_K(cf)^γ_K

≥

3

X

i=1 k+1

X

β=1

˜ wβ

1

6r_K^i,β− 4t(fP)^i,β_K r^i,β_K φ⁻¹_?

+

L

X

γ=1

ˆ wγ

1

6r_K^γ − 4t(fP)^γ_Kr_K^γφ⁻¹_?

=

3

X

i=1 k+1

X

β=1

˜ wβ

1

6 − 4t(fP)^i,β_K φ⁻¹_?

r_K^i,β+

L

X

γ=1

ˆ wγ

1

6 − 4t(fP)^γ_Kφ⁻¹_?

r^γ_K. ThusL₂>0under the condition (4.11).

L3=1

6r¯_Kⁿ − 4tB1(φ)cf

=1 6





3

X

i=1 k+1

X

β=1

˜ wβr^i,β_K +

L

X

γ=1

ˆ wγr^γ_K



− 4t





3

X

i=1 k+1

X

β=1

˜

wβB1(φ^i,β_K )(cf)^i,β_K +

L

X

γ=1

ˆ

wγB1(φ^γ_K)(cf)^γ_K





=

3

X

i=1 k+1

X

β=1

˜ wβ

1

6r_K^i,β− 4tB1(φ^i,β_K )(cf)^i,β_K

+

L

X

γ=1

ˆ wγ

1

6r_K^γ − 4tB1(φ^γ_K)(cf)^γ_K

≥

3

X

i=1 k+1

X

β=1

˜ wβ

1

6r_K^i,β− 4tB1(φ^i,β_K )r^i,β_K φ⁻¹_?

+

L

X

γ=1

ˆ wγ

1

6r_K^γ − 4tB1(φ^γ_K)r^γ_Kφ⁻¹_?

=

3

X

i=1 k+1

X

β=1

˜ wβ

1

6 − 4tB1(φ^i,β_K )φ⁻¹_?

r_K^i,β+

L

X

γ=1

ˆ wγ

1

6 − 4tB1(φ^γ_K)φ⁻¹_?

r_K^γ

≥

3

X

i=1 k+1

X

β=1

˜ wβ

1

6 − 4tB1(φ?)φ⁻¹_?

r_K^i,β+

L

X

γ=1

ˆ wγ

1

6 − 4tB1(φ?)φ⁻¹_?

r^γ_K.

ThusL₃ >0 under the condition (4.11). To sum up,H_K^s(r, c_f, c_I, f_P, f_I, φ) =L₁+L₂+L₃ >0 under the condition (4.11).

In the following two lemmas, we only consider second order scheme, i.e. we use P¹ element, and apply quadrature rule in [28] withk = 1 in cell and the the corresponding2 point Gaussian quadrature rule on cell interface. Note that in this case,ωˆβ= ¹₃ωβ.

Lemma 4.3. Given rⁿ>0 (cⁿ_f >0), we have H_K^c (r, c_f,u)>0, ifαsatises α≥max

i,β,K{|u^i,β_K|}, (4.12)

and the time step satises

4t≤ min

i,β,m{ |K|φ(Vm)

9`ⁱ_K(|u^i,β_K |+α)}, (4.13)

whereφ(Vm),m= 1,2,3 are the values ofφat verticesVm∈K at time leveln. Proof. Applying quadrature rule fork= 1, we can rewrite (4.8) as

H_K^c = 1 3r¯ⁿ_K−λ

3

X

i=1

Z

eⁱ_Kucdf·ν_Kⁱ ds

= 1 9





3

X

i=1 2

X

β=1

wβr^i,β_K



−λ

3

X

i=1 2

X

β=1

wβ`ⁱ_K(ucdf)^i,β_ei K

·ν_Kⁱ

=

3

X

i=1 2

X

β=1

wβ

1

9r^i,β_K −λ`ⁱ_K(ucdf)^i,β_ei K

·ν_Kⁱ

=

3

X

i=1 2

X

β=1

w_β 1

9r^i,β_K −λ`ⁱ_K 1

2u^i,β_K

i·ν_Kⁱ (c_f)^i,β_K

i+1

2u^i,β_K ·ν_Kⁱ (c_f)^i,β_K −α(c_f)^i,β_K

i+α(c_f)^i,β_K

=

3

X

i=1 2

X

β=1

wβ

( 1

18r_K^i,β−1 2λ`ⁱ_K

u^i,β_K

i·ν_Kⁱ (cf)^i,β_K

i−α(cf)^i,β_K

i+α(cf)^i,β_K +

1

18r^i,β_K −1 2λ`ⁱ_K

u^i,β_K ·ν_Kⁱ (c_f)^i,β_K −α(c_f)^i,β_K

i+α(c_f)^i,β_K ) :=

3

X

i=1 2

X

β=1

wβ

L^i,β₁ +L^i,β₂ .

Sincec_f andrare both approximated by linear functions, they can be represented as a linear combination of their values on three vertices{V1, V2, V3} ofK, i.e. for any pointx^ρ_K ∈K,

(c_f)^ρ_K=

3

X

m=1

µ^ρ_mc_f(V_m), r^ρ_K =

3

X

m=1

µ^ρ_mr(V_m) =

3

X

m=1

µ^ρ_mφ(V_m)c_f(V_m), (4.14) where0≤µ^ρ₁, µ^ρ₂, µ^ρ₃≤1andµ^ρ₁+µ^ρ₂+µ^ρ₃= 1are the barycentric coordinates ofx^ρ_K in K. Then we have

L^i,β₁ = 1

18r_K^i,β−1 2λ`ⁱ_K

u^i,β_K

i·ν_Kⁱ (cf)^i,β_K

i−α(cf)^i,β_K

i+α(cf)^i,β_K

=

3

X

m=1

1

18µ^i,β_mφ(Vm)cf(Vm)−1

2λ`ⁱ_K u^i,β_K

i·ν_Kⁱ (cf)^i,β_K

i−α(cf)^i,β_K

i+α

3

X

m=1

µ^i,β_mcf(Vm)

!

=

3

X

m=1

µ^i,β_m 1

18φ(Vm)−1 2λ`ⁱ_Kα

cf(Vm) +1

2λ`ⁱ_K(α−u^i,β_K

i·ν_Kⁱ )(cf)^i,β_K

i,

and

L^i,β₂ = 1

18r^i,β_K −1 2λ`ⁱ_K

u^i,β_K ·ν_Kⁱ (cf)^i,β_K −α(cf)^i,β_K

i +α(cf)^i,β_K

=

3

X

m=1

1

18µ^i,β_mφ(Vm)cf(Vm)−1

2λ`ⁱ_K u^i,β_K ·ν_Kⁱ

3

X

m=1

µ^i,β_mcf(Vm)−α(cf)^i,β_K

i+α

3

X

m=1

µ^i,β_mcf(Vm)

!

=

3

X

m=1

µ^i,β_m 1

18φ(Vm)−1

2λ`ⁱ_K(u^i,β_K ·ν_Kⁱ +α)

cf(Vm) +1

2λ`ⁱ_Kα(cf)^i,β_K

i.

ThereforeL^i,β₁ , L^i,β₂ >0under the conditions (4.12) and (4.13), respectively, which further yieldsH_K^c >0. Lemma 4.4. Given rⁿ>0 (cⁿ_f >0), we haveH_K^d(r, c_f,u, φ)>0, ifα˜ satises

˜

α≥ (3 +√ 3)Λ

2ρ , (4.15)

and the time step satises

∆t≤min

m {|K|φ(Vm)

18 ˜α }, ∆t≤min

m { ρ|K|φ(Vm) 27(3 +√

3)Λ}, (4.16)

whereφ(V_m),m= 1,2,3are the values ofφ at the verticesV_m∈K at time level n,ρis the parameter used in the denition of regularity of Ωh, andΛ is the largest spectral radius ofD inK's.

Proof. For the diusion part

H_K^d(r, cf,u, φ) = 1 3r¯ⁿ_K+λ

3

X

i=1

Z

eⁱ_K

{D(u)∇cf} ·ν_Kⁱ + α˜

`ⁱ_K[cf]ne·ν_Kⁱ

ds.

SinceDis symmetric, following [32], we can rewrite the diusion term as a directional derivative D∇c_f·ν_Kⁱ =Dν_Kⁱ · ∇c_f =Sⁱ∂cf

∂lⁱ,

where Sⁱ = kDν_Kⁱ k ≤ Λ and lⁱ = Dν_Kⁱ /kDν_Kⁱ k. Dene S_Kⁱ = Sⁱ|_K, S_Kⁱ

i = Sⁱ|_Ki and lⁱ_K = lⁱ|_K, lⁱ_K

i =lⁱ|K_i. For each quadrature pointx^i,β_K on the edgeeⁱ_K, we can nd the corresponding pointsx˜^i,β_K ∈∂K and x˜^i,β_K

i ∈∂Ki such that −−−−−→

˜

x^i,β_K x^i,β_K and −−−−−→ x^i,β_K x˜^i,β_K

i are the same direction with lⁱ_K and lⁱ_K

i, respectively. See Figure 2 for an illustration. At the quadrature pointx=x^i,β_K , we have

{D(u)∇cf}^i,β_ei K

·ν_Kⁱ =1

2D(u^i,β_K )∇(cf)^i,β_K ·ν_Kⁱ +1 2D(u^i,β_K

i)∇(cf)^i,β_K

i·ν_Kⁱ

=1

2S_K^i,β(c_f)^i,β_K −c_f(˜x^i,β_K ) kx^i,β_K −x˜^i,β_K k +1

2S_K^i,β

i

cf(˜x^i,β_K

i)−(cf)^i,β_K

i

kx˜^i,β_K

i −x^i,β_K k

= S_K^i,β

2kx^i,β_K −x˜^i,β_Kk(c_f)^i,β_K − S_K^i,β

i

2k˜x^i,β_K

i−x^i,β_Kk(c_f)^i,β_K

i− S_K^i,β

2kx^i,β_K −x˜^i,β_Kkc_f(˜x^i,β_K ) + S_K^i,β

i

2kx˜^i,β_K

i −x^i,β_K kc_f(˜x^i,β_K

i).

K Ki

•

˜ x^i,β_K

• x^i,β_K

•

˜ x^i,β_K

i

eⁱ_K ν_Kⁱ

Figure 2: The points chosen to evaluate directional derivative in the diusion part.

Therefore, we can rewriteH_K^d(r, cf,u, φ)as

H_K^d = 1 6r¯ⁿ_K+1

6¯r_Kⁿ +λ

3

X

i=1

Z

eⁱ_K

{D(u)∇cf} ·ν_Kⁱ + α˜

`ⁱ_K[cf]ne·ν_Kⁱ

ds

= 1

6r¯ⁿ_K+ 1 18

3

X

i=1 2

X

β=1

wβr^i,β_K +λ

3

X

i=1 2

X

β=1

wβ`ⁱ_K

{D(u)∇cf}^i,β_ei K

·ν_Kⁱ + α˜

`ⁱ_K(cf)^i,β_K

i− α˜

`ⁱ_K(cf)^i,β_K

= 1

6r¯ⁿ_K+ 1 18

3

X

i=1 2

X

β=1 3

X

m=1

wβµ^i,β_mφ(Vm)cf(Vm) +λ

3

X

i=1 2

X

β=1

wβ`ⁱ_K

{D(u)∇cf}^i,β_ei K

·ν_Kⁱ + α˜

`ⁱ_K(cf)^i,β_K

i− α˜

`ⁱ_K(cf)^i,β_K

= 1 6r¯ⁿ_K+

3

X

i=1 2

X

β=1

w_β 1 18

3

X

m=1

µ^i,β_mφ(V_m)c_f(V_m) +λ`ⁱ_K

{D(u)∇c_f}^i,β_ei

K

·ν_Kⁱ + α˜

`ⁱ_K(c_f)^i,β_K

i− α˜

`ⁱ_K(c_f)^i,β_K !

:=

3

X

i=1 2

X

β=1

wβL^i,β₁ +L2, where

L^i,β₁ = 1 18

3

X

m=1

µ^i,β_mφ(V_m)c_f(V_m) +λ`ⁱ_K

"

S_K^i,β

2kx^i,β_K −x˜^i,β_Kk − α˜

`ⁱ_K

!

(c_f)^i,β_K + α˜

`ⁱ_K − S_K^i,β

i

2kx˜^i,β_K

i−x^i,β_K k

! (c_f)^i,β_K

i

+ S_K^i,β

i

2kx˜^i,β_K

i−x^i,β_K kc_f(˜x^i,β_K

i)

# ,

L2= 1

6r¯ⁿ_K−λ`ⁱ_K

3

X

i=1 2

X

β=1

ωβS_K^i,β

2kx^i,β_K −x˜^i,β_K kcf(˜x^i,β_K ).

h BK_i

Ki

s

θ^j_K

i

Figure 3: TriangleK_i and its sine

We need to makeL^i,β₁ , L₂>0. In fact

L^i,β₁ = 1 18

3

X

m=1

µ^i,β_mφ(Vm)cf(Vm) +λ`ⁱ_K S_K^i,β

2kx^i,β_K −x˜^i,β_Kk − α˜

`ⁱ_K

! (cf)^i,β_K

+λ`ⁱ_K α˜

`ⁱ_K − S_K^i,β

i

2k˜x^i,β_K

i−x^i,β_Kk

! (cf)^i,β_K

i+λ`ⁱ_K S_K^i,β

i

2k˜x^i,β_K

i−x^i,β_K kcf(˜x^i,β_K

i)

=

3

X

m=1

µ^i,β_m 1

18φ(V_m) +λ`ⁱ_K S_K^i,β

2kx^i,β_K −x˜^i,β_K k− α˜

`ⁱ_K

!!

c_f(V_m)

+λ`ⁱ_K α˜

`ⁱ_K − S_K^i,β

i

2k˜x^i,β_K

i−x^i,β_Kk

! (cf)^i,β_K

i+λ`ⁱ_K S_K^i,β

i

2k˜x^i,β_K

i−x^i,β_K kcf(˜x^i,β_K

i).

SinceS_K^i,β, S_K^i,β

i ≤Λ, to makeL^i,β₁ >0, we need

˜

α≥ `ⁱ_KΛ 2k˜x^i,β_K

i−x^i,β_K k, λ`ⁱ_K α˜

`ⁱ_K − S_K^i,β 2kx^i,β_K −x˜^i,β_K k

!

≤ 1 18φ(Vm).

It's easy to compute that

`ⁱ_K k˜x^i,β_K

i−x^i,β_K k ≤ 3 +√ 3 minjsin

θ^j_K

i

, where the θ^j_K

i is the angle in triangle Ki which is opposite to the edgee^j_K

i. From Figure 3 and regularity assumption ofΩ_h, for all angleθ_K^j

i inK_i, we have sinθ^j_K

i =h

s ≥diam(BK_i) diam(K_i) ≥ρ.

ThereforeL^i,β₁ >0under the conditions (4.15) and (4.16). As forL2, similar to (4.14), we write

c_f(˜x^i,β_K ) =

3

X

m=1

˜

µ^i,β_mc_f(V_m),