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Numerical approximation for a Baer–Nunziato model of two-phase flows

Mai Duc Thanha, ∗ , Dietmar Krönerb, Nguyen Thanh Namc

aDepartment of Mathematics, International University, Quarter 6, Linh Trung Ward, Thu Duc District, Ho Chi Minh City, Viet Nam

bInstitute of Applied Mathematics, University of Freiburg, Hermann-Herder Str 10, 79104 Freiburg, Germany

cNational Key Laboratory of Digital Control and System Engineering, Block C6, 268 Ly Thuong Kiet street, Ward 14, District 10, Ho Chi Minh City, Viet Nam

Article history:

Received 5 February 2009

Received in revised form 13 January 2011

Accepted 13 January 2011

Available online 18 January 2011

Keywords:

Two-phase flow

Conservation law

Source term

Numerical approximation

Lax–Friedrichs

Well-balanced scheme

We present a well-balanced numerical scheme for approximating the solution of the Baer–Nunziato model of two-phase flows by balancing the source terms and discretizing the compaction dynamics equation First, the system is transformed into a new one of three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation law of the mass in the solid phase and the conservation law of the momentum of the mixture, and the compaction dynamic equation

is considered as the third subsystem In the first subsystem, stationary waves are used

to build up a well-balanced scheme which can capture equilibrium states The second subsystem is of conservative form and thus can be numerically treated in a standard way For the third subsystem, the fact that the solid velocity is constant across the solid contact suggests us to compose the technique of the Engquist–Osher scheme We show that our

scheme is capable of capturing exactly equilibrium states Moreover, numerical tests show

the convergence of approximate solutions to the exact solution

©2011 IMACS Published by Elsevier B.V All rights reserved

1 Introduction

We are interested in numerical approximations for the solutions of the Baer–Nunziato (BN) model of two-phase flows which was introduced by Baer and Nunziato [4] for the study of the deflagration-to-detonation transition (DDT) in granular explosives This two-phase treatment of the explosive as a mixture is mathematically formulated in terms of variables for its two separate constituents: a granular solid phase and a separate combustion product gas phase The mixture is assumed

to be immiscible but the two phases are not in equilibrium with each other Each phase is identifiable and characterized by

a separate equation of state (EOS) The volume fraction of each phase is a dependent variable required to specify the state

of the mixture Each phase satisfies the balance laws of mass, momentum and energy Interactions between two phases are described by source terms for the exchange of mass, momentum and energy between the phases The model is enclosed by the compaction dynamics equation which describes the evolution of the volume fraction variable This equation indicates the way in which microstructural forces at the interphases act to derive the volume fraction toward equilibrium states See [8] for a crucial review and the references therein for the modeling, analysis and numerical simulation of DDT in porous energetic materials The governing equations of the BN model are in accordance with the general framework [11] which formulates the mathematical derivation of multi-phase flow models See also [29,12], etc., for the modeling of two-pressure two-phase flows

In this paper, we consider the BN model in the nonreactive and isentropic case Furthermore, the interfacial pressure is assumed to be equal to the pressure in the gas phase, and the mass and the residual momentum exchanges between phases

*Corresponding author.

E-mail addresses:mdthanh@hcmiu.edu.vn (M.D Thanh), dietmar@mathematik.uni-freiburg.de (D Kröner), thanhnam@dcselab.edu.vn (N.T Nam) 0168-9274/$30.00©2011 IMACS Published by Elsevier B.V All rights reserved.

are neglected Precisely, the model is described by a system of four equations characterizing the conservation of mass and the balance of momentum in each phase:

∂t( αgρg) + ∂x( αgρg u g) =0,

∂t( αgρg u g) + ∂x



αg



ρg u2g+p g



=p g∂xαg,

∂t( αsρs) + ∂x( αsρs u s) =0,

∂t( αsρs u s) + ∂x



αs



ρs u2s+p s

together with the compaction dynamics equation

Throughout, we use the subscripts g and s to indicate the quantities in the gas phase and in the solid phase, respectively.

The notations αk, ρk , u k , p k , k=g,s, respectively, stand for the volume fraction, density, velocity, and pressure in the

k-phase, k=g,s The volume fractions satisfy

Each phase has an equation of state of the form

p k= κkργ k

The system (1.1)–(1.2) has the form of nonconservative systems of balance laws In nonconservative systems, there are often a part of conservative terms and a part of nonconservative terms Without nonconservative terms, the systems can

be well dealt with by the standard theory of hyperbolic systems of conservations laws Nonconservative terms such as the

terms p g∂xαg,u s∂xαg in the above model or source terms in other models (see [24,34,36] for instance) in general involve the product of a discontinuous function and the (partial) derivative of another quantity that might be discontinuous as well Mathematical formulation of nonconservative systems of balance laws was introduced in [10] The study of the impact

of nonconservative terms plays a key role in nonconservative systems and, in particular, in multi-phase flows models In general, nonconservative terms may cause the ill-posedness for the boundary/initial-value problem, as there can be multiple solutions, see [26,34], for example There have been many contributions for the study of wave structures and the Riemann problem for various models of nonconservative systems An early research for the model of a fluid in a nozzle with discon-tinuous cross-section was carried out in [24] This was followed by a sequence of papers [26,34] for the same model, [27,36] for shallow water equations, and [13] for the case of a general system In [34], the uniqueness of the Riemann solutions of the model of a fluid in a nozzle with piece-wise cross-section has been established In [2,32], exact Riemann solutions of the BN model were constructed See also the references therein for related works Numerically, the nonconservative terms often cause lots of inconveniences in approximating physical solutions of the system This has been seen even in the case

of a single conservation law with a source term (see [6] for example) The discretization of nonconservative terms therefore has been an attractive topic for many years Basically, a good numerical method for a nonconservative system should give

a good approximation of the exact solution in the typical case where the system is stable and stationary (independent of time) whenever such a situation exists This case corresponds to the equilibrium states which can be obtained using sta-tionary waves This has motivated, particularly, the study of well-balanced schemes for nonconservative systems that can capture equilibrium states, see [6,21,36,18,19,37] for equilibrium sate capturing schemes for various models

Many nonconservative systems, such as the ones we just talked about above, possess the characteristic property that: all

the eigenvalues of the Jacobian matrix are real and given in explicit forms, and that the characteristic fields may coincide

on certain surfaces of the phase domain This is a very interesting phenomenon in the sense that on one hand the non-conservativeness poses challenging problems, on the other hand the fact that all the characteristic fields are real and explicit raises the hope for the study in the framework of hyperbolic systems Observe that many one-pressure two-fluid models, where pressures on both phases are assumed to be equilibrium with each other, do not possess this property, see [33] As seen later on, the system (1.1)–(1.2) is hyperbolic as all the characteristic fields are real and explicitly given Moreover, it is not strictly hyperbolic as some characteristic fields may coincide on certain surfaces of the phase domain

Our goal in this paper is to provide a reliable numerical method for approximating the solution of the system (1.1)–(1.2) that can capture equilibrium states First, to reduce the nonconservative terms, we transform the system into a new one which is combined from three subsystems: the first subsystem consists of the balance laws in the gas phase, the second subsystem consists of the conservation laws of the mixture, and the compaction dynamic equation represents the third subsystem Observe that each of these three subsystems is different from each other in the nature and thus will be treated separately and differently In the first subsystem, stationary waves are used to build up a well-balanced scheme which can capture equilibrium states This explain why our method can capture exactly equilibrium states The second subsystem

is numerically treated in a usual way as it is conservative For the third subsystem, we observe that the solid velocity is constant across the solid contact and that the compaction dynamics equation becomes conservative in the region where the velocity of the solid phase is constant This suggests us to employ the technique of the Engquist–Osher scheme to treat the

compaction dynamics equation We show that our scheme capture exactly the equilibrium states and numerical tests provide us

with reasonable approximations of the exact Riemann solutions of the model

There have been many contributions devoted to the discretization of the nonconservative terms in nonconservative sys-tems of balance laws In the case of a single conservation law with a source term, numerical well-balanced schemes were presented in [14,15,6,5,3] A well-balanced scheme for the model of fluid flows in a nozzle with variable cross-section was proposed in [21] and its properties are studied in [22] Well-balanced schemes for one-dimensional shallow water equa-tions were constructed in [36,9] In [31] the authors propose to take into account the nonconservative terms using the free streaming physical condition with uniform velocity and pressure profiles See also [7,23,30] and the references therein for related works A well-balanced scheme for a one-pressure model of two-phase flows where one phase is compressible, the other phase is incompressible was constructed in [35], where the impact of stationary waves is required to take place on the full system In [28], the author takes the comparison of Roe-type methods for solving a two-fluid model with and without pressure relaxation Recently, a relaxation and numerical approximation were presented in [1], and a hybrid scheme was presented in [20] for the Baer–Nunziato model

The outline of this paper is as follows In Section 2 we provide basic properties of the system (1.1)–(1.2) Section 3 deals with stationary waves In Section 4, we construct the numerical method for the BN model Section 5 is devoted to numerical tests

2 Preliminaries

2.1 Hyperbolicity

For smooth solutions, the system (1.1)–(1.2) is equivalent to the following system

∂tρg+u g∂xρg+ ρg∂x u g=0,

∂t u g+h

g( ρg)∂xρg+u g∂x u g=0,

∂tρs+u s∂xρs+ ρs∂x u s=0,

∂t u s+h

s( ρs)∂xρs+u s∂x u s+ p g−p s

(1− αg) ρs

∂xαg=0,

where h k is the specific enthalpy of the k-phase:

h

k( ρ ) = pk( ρ )

ρ , k=s g.

From (2.1), choosing the dependent variable U= ( ρg,u g, ρs,u s, αg), we can rewrite the system (1.1)–(1.2) as a system of balance laws in nonconservative form as

where

A(U) =

⎛

⎜

⎜

⎝

u g ρg 0 0 0

h

g( ρg) u g 0 0 0

s( ρs) u s p g−p s

(1−α g ) ρ s

⎞

⎟

⎟

⎠ .

The characteristic equation is given by

(u s− λ)  (u g− λ)2−p

g



(u s− λ)2−p

s



=0,

which admits five roots as

λ1(U) =u g− p

g, λ2(U) =u g+ p

g,

λ3(U) =u s− p

s, λ4(U) =u s+ p

Thus, the system is hyperbolic and the corresponding right eigenvectors can be chosen as

r1(U) = −2 p



g( ρg)

p

g( ρg) ρg+2p

g( ρg)



ρg, − p

g,0,0,0T

,

r2(U) = 2 p



g( ρg)

p

g( ρg) ρg+2p

g( ρg)



ρg, p

g,0,0,0T

,

r3(U) = −2

p

s( ρs)

p

s( ρs) ρs+2p

s( ρs)



0,0, ρs, − p

s,0T ,

r4(U) = 2

p

s( ρs)

p

s( ρs) ρs+2p

s( ρs)



0,0, ρs,

p

s,0T ,

r5(U) =

0,0,p s( ρs) −p g( ρg)

(1− αg) ρs

,0,h

s( ρs)

T

It is not difficult to verify that

Dλi(U) ·r i(U) =1, i=1,2,3,4,

so that the first, second, third, fourth characteristic fields (λi(U),r i(U)), i=1,2,3,4, are genuinely nonlinear, while the fifth characteristic field(λ5(U),r5(U))is linearly degenerate

Observe that the characteristic fields(λ1(U),r1(U)) and(λ2(U),r2(U))may coincide with any remaining characteristic fields(λ3(U),r3(U)),(λ4(U),r4(U))and(λ5(U),r5(U))on certain surfaces Thus, the system is not strictly hyperbolic in the whole domain

2.2 Rarefaction waves

Next, let us look for rarefaction waves of the system (2.2), i.e., the continuous piecewise-smooth self-similar solutions of the form

U(x,t) =V(ξ ), ξ =x

t,t>0,x∈ R.

Substituting this into (2.2), we can see that rarefaction waves are solutions of the following initial-value problem for ordinary differential equations

dV(ξ )

dξ =r i

V(ξ ) 

, ξ  λi



U0

,i=1,2,3,4,

V

λi



U0

For the first characteristic field, (2.6) yields

dρg(ξ )

dξ = −2 p



g( ρg)

p

g( ρg) ρg+2p

g( ρg) ρg(ξ ) <0,

du g(ξ )

dξ = 2 p



g( ρg)

p

g( ρg) ρg+2p

g( ρg) p



g(ξ ) >0,

dρs(ξ )

dξ =du s(ξ )

dξ =dαg(ξ )

This implies thatρs,u s, αg are constant through 1-rarefaction waves,ρgis decreasing with respect toξand u gis increasing with respect toξ Moreover, u gcan be resolved fromρgby

du g

dρg = − p



g( ρg)

ρg

which determines the curve R1(U0) consisting of all right-hand states that can be connected to the left-hand state U0 using 1-rarefaction waves The trajectory u =u ( ρ )of (2.8) starting at U is given by



U0

: u g=u0g−

ρ g



ρ0

p

g(y)

y dy, ρg ρ0

since the characteristic speed must be increasing through a rarefaction fan

Argue similarly, we can see that ρs , u s,αg are constant through 2-rarefaction waves, and the second rarefaction curve

R2(U0)starting from U0 is given by

R2



U0

: u g=u0g+

ρ g



ρ0

p

g(y)

y dy, ρg ρ0

Through 3-rarefaction fans,ρg,u g, αg are constant, and the third rarefaction curveR3(U0)starting from U0is given by

R3



U0

: u s=u0s−

ρ s



ρ0

s

p

s(y)

Through 4-rarefaction fans,ρg , u g,αg are constant, and the fourth rarefaction curveR4(U0)starting from U0 is given by

R4



U0

: u s=u0s+

ρ s



ρ0

s

p

s(y)

2.3 Shock waves and contact discontinuities

It has been shown that, see [2,32] for example, along a discontinuity:

(i) either the volume fractions are constant;

(ii) or u s≡constant and the discontinuity propagates with a constant speedλ =u s

In the case (i), the system then becomes two independent subsystems of isentropic gas dynamics equations in each

phase Given a left-hand state U0, the shock curvesSi(U0), i=1,2,3,4 consisting of all right-hand states U that can be connected to U0 by a Lax shock are given by:

S1,3(U0): u k=u k0−

κk

1

ρk0− 1

ρk

ρk γ − ρk0 γ

1/2

, ρk> ρk0,

S2,4(U0): u k=u k0−

κk

1

ρk0− 1

ρk ρ

γ

k − ρk0 γ

1/2

The four wave curves in nonlinear characteristic families are then defined by:

Wi



U0

:= Ri



U0

∪ Si



U0

, i=1,2,3,4.

It is easy to see that along the curvesW1(U0), W3(U0), the velocity is a monotone decreasing function of the density; along the curvesW2(U0),W4(U0), the velocity is a monotone increasing function of the density

In the case (ii), the discontinuity satisfies the following jump relations



αgρg(u g−u s) 

=0,

u2

g

2 −u g u s+h g



=0,



αgρg(u g−u s)u g+ αg p g+ αs p s

Thus, the left-hand state U0 and right-hand state U of a contact discontinuity associated with the characteristic speed

λ5=u s≡constant satisfy

αgρg(u g−u s) = αg0ρg0(u g0−u s) :=m,

(u g−u s)2+2h g= (u g0−u s)2+2h g0,

The above argument leads to defining elementary waves of the system (1.1)–(1.2), which make up solutions of the Riemann problem

Definition 2.1 (Elementary waves) The elementary waves of the system (1.1)–(1.2) are

(i) Rarefaction waves these waves are continuous piecewise-smooth self-similar solutions of the system (1.1)–(1.2); (ii) Lax shocks in each phase where the system becomes the usual isentropic gas dynamics equations;

(iii) Contact discontinuities associated with the linearly degenerate characteristic field λ5=u s where the left-hand and right-hand states are defined by (2.15)

3 Stationary contacts in the gas phase

As observed earlier, nonconservative terms often cause lots of inconveniences for numerical approximations To reduce the number nonconservative terms, we add up the two equations of balance of momentum to get the conservation of momentum of the total in place of the equation of balance of momentum for the liquid phase So we get three sets of equations:

– governing equations in the gas phase:

∂t( αgρg) + ∂x( αgρg u g) =0,

∂t( αgρg u g) + ∂x



αg



ρg u2g+p g

– “composite” conservation laws:

∂t( αsρs) + ∂x( αsρs u s) =0,

∂t( αsρs u s+ αgρg u g) + ∂x



αs



ρs u2s+p s

+ αg



ρg u2g+p g

– compaction dynamics equation:

Set the dependent conservative variable

V= ( αgρg, αgρg u g, αsρs, αgρg u g+ αsρs u s)T,

the flux

f(V) =  αgρg u g, αg



ρg u2g+p g

, αsρs u s, αs



ρs u2s+p s

+ αg



ρg u2g+p gT

,

and the source

S(V) = (0,p g∂xαg,0,0)T.

We can see that a unique source appears only in the second component Thus, we can rewrite the system (3.1)–(3.2) as a

system of conservation laws with a single source term

∂t V(x,t) + ∂x f

V(x,t) 

=S

V(x,t) 

Observe that the system (3.4) is under-determined as the number of unknowns is larger than the number of equations We still need the compaction dynamics equation (3.3) for the closure of the system

We need to study stationary contacts of the system (3.4) which takes into account the nonconservative terms Therefore,

we need only to consider the governing equations (3.1) of the gas phase

To simplify the expressions, in the rest of this section, we omit the subscript g in the gas phase Motivated by our earlier

works [26,21], we look for stationary contacts that are the limit of stationary smooth solutions of (3.1) A stationary smooth

solution U of (3.1) is a time-independent smooth solution so that it satisfies the following ordinary differential equations

( αρu)=0,

u2

2 +h



where(.)=d/dx and h( ρ ) =p( ρ )/ ρ ,or

h( ρ ) = κγ

γ −1ργ−1.

Arguing similarly as in [26,21], we obtain the following conclusion

Lemma 3.1 The left-hand and right-hand states U±of a stationary contact for (3.1) satisfy

[ αρu] =0,



u2

2 +h



where[ αρu] := α ρ u+− α ρ u−, and so on, denotes the difference of the corresponding valueαρu between the right-hand and

left-hand states of the stationary contact.

From Lemma 3.1, we deduce that a stationary wave of (3.1) from a given state U0= ( α0, ρ0,u0) to some state U =

( α , ρ ,u)must satisfy the relations

αρu= α0ρ0u0,

u2

2 +h( ρ ) =u20

It is derived from (3.7) that the density is a root of the nonlinear algebraic equation

F(U0, ρ , α ) :=sgn(u0)

u20− 2κγ

γ −1



ργ−1− ρ0γ−1

1/2

ρ − α0u0ρ0

To find zeros of the function F(U0, ρ , α ), we need to investigate its properties Observe that the function F(U0, ρ , α ) is well-defined whenever

u20− 2κγ

γ −1



ργ−1− ρ0γ−1

0,

or

ρ  ¯ ρ (U0) :=

γ −1

2κγ u

2

0+ ρ0γ−1

1

γ− 1

.

We have

∂ (U0, ρ ; α )

∂ ρ =u

2−2κγ

γ−1( ργ−1− ρ0γ−1) − κγργ−1



u20−2κγ

γ−1( ργ−1− ρ0γ−1) 1/2 .

Assume, for definitiveness, that u0>0 The last expression yields

∂ (U0, ρ ; α )

∂ ρ >0, ρ < ρmax( ρ0,u0),

∂ (U0, ρ ; α )

∂ ρ <0, ρ > ρmax( ρ0,u0),

where

ρmax( ρ0,u0) :=

γ −1

κγ ( γ +1)u

2+ 2

γ +1ρ0γ−1

1

γ− 1

Since

F(U0, ρ =0,a) =F(U0, ρ = ¯ ρ ,a) = − α0u0ρ0

α <0,

the functionρ →F(U0, ρ ; α )admits a root if and only if the maximum value is nonnegative:

F(U0, ρ = ρmax, α ) 0,

or, equivalently,

α  αmin(U0) := α0ρ0|u0|

√

κγ ρ

γ+ 1 max( ρ0,u0)

Similar argument can be made for u0<0

It will be convenient to define in the ( ρ ,u)-plan the following sets, referred to as the “lower region” G1, the “middle

region” G , and the “upper region” G , and the “boundary”C, as

Fig 1 The regions G1, G2, G3 and the boundaryCin the( ρ , u )-plane.

G1:=  ( ρ ,u): u< − p( ρ ) 

,

G2:=  ( ρ ,u):|u| < p( ρ ) 

,

G3:=  ( ρ ,u): u>

p( ρ ) 

,

C :=  ( ρ ,u): u= ± p( ρ ) 

see Fig 1

The above argument leads us to the following results

Lemma 3.2 Given U0= ( α0, ρ0,u0)and 0 α 1 The function F(U0, ρ , α )in (3.8) admits a zero if and only if a αmin(U0) In this case, F(U0, ρ , α )admits two distinct zeros, denoted byρ = ϕ1(U0, α ), ρ = ϕ2(U0, α )such that

the equality in (3.12) holds only ifα = αmin(U0).

Lemma 3.3.

(i) Letρmax( ρ0,u0)be defined as (3.9) Then, it holds that

ρmax( ρ0,u0) < ρ0, ( ρ0,u0) ∈G2,

ρmax( ρ0,u0) > ρ0, ( ρ0,u0) ∈G3∪G1,

(ii) Given U0= ( α0, ρ0,u0)and 0 α 1 Let u be defined by (3.7) and letϕi(U0, α ), i=1,2,be defined in Lemma 3.2 The state ( ϕ1(U0, α ),u) ∈G1if u0<0, and the state( ϕ1(U0, α ),u) ∈G3if u0>0; the state( ϕ2(U0, α ),u) ∈G2 Moreover,



ρmax( ρ0,u0),u0



In addition, we have

– Ifα > α0, then

ϕ1(U0, α ) < ρ0< ϕ2(U0, α ). (3.15)

– Ifα < α0, then

ρ0< ϕ1(U0, α ) for( ρ0,u0) ∈G1∪G3,

(iii) Given U= ( α , ρ ,u)and letαmin(U)be defined as in (3.10) The following conclusions hold

αmin(U) < α , ( ρ ,u) ∈G i,i=1,2,3,

αmin(U) = α , ( ρ ,u) ∈ C ,

Proof Most of the proof was available in [26] However, for completeness, we will show the steps Assume for simplicity

that u0>0 Define

g(U0, ρ ) =u20− 2κγ

γ −1



ργ−1− ρ0γ−1

Then, a straightforward calculation gives

g

U0, ρmax(U0) 

=0,

which proves (3.14) On the other hand, since

dg(U0, ρ )

dρ = −( γ +1) κγργ−2<0,

and thatϕ1(U0, α ) < ρmax(U0, α ) < ϕ2(U0, α )it holds that

g

U0, ϕ1(U0, α ) 

>g

U0, ρmax(U0) 

=0>g

U0, ϕ1(U0, α ) 

.

The last two inequalities justify the statement in (b) Moreover,

F(U0, ρ0; α ) = ρ0u0(1− α0/ α ) >0 iff a> α0,

which proves (3.15), and shows that ρ0is located outside of the interval[ ϕ1(U0, α ), ϕ2(U0, α ) ]in the opposite case Since

∂ (U0, ρ0; α )

∂ ρ =u20− κγρ0γ−1

u0 <0 iff U0∈G2,

which, together with the earlier observation, implies (3.16)

We next check (3.17) for a= α0 It comes from the definition ofαmin(U0)thatαmin(U0) < α0if and only if

√

κγ ρ∗γ+1 > ρ0|u0|,

that can be equivalently written as

Q(m) := 2

γ +1m− ( κγ )

1 −γ

γ+ 1m γ2+ 1 + γ −1

κγ ( γ +1) >0,

where m:= ργ0−1/u2 Then, we can see that

which, in particular shows that the second equation in (3.17) holds, since( ρ0,u0) ∈ C±for m=1/ κγ Moreover,

d Q(m)

dm = 2

γ +1



1− ( κγm)

1 −γ

γ+ 1

,

which is positive for m>1/ κγ and negative for m< κγ This together with (3.19) establish the first statement in (3.17) The third statement in (3.17) is straightforward This completes the proof of Lemma 3.3 2

To select a unique stationary wave, we need the following so-called Monotonicity Criterion The relationships (3.6) also define a curveρ → α = α (U0, ρ ) So we require that

Monotonicity Criterion The volume fraction α = α (U0, ρ )must vary monotonically between the two values ρ0 andρ1, whereρ1 is theρ-value of the corresponding state of a stationary wave having U0 as one state

A similar criterion was used by Kröner, LeFloch, and Thanh [25,21,22], Isaacson and Temple [16,17]

Geometrically, we can choose eitherϕ1orϕ2 in the domains G1, G2, G3 using the following lemma

Lemma 3.4 The Monotonicity Criterion is equivalent to saying that any stationary shock does not cross the boundaryC In other words:

(i) If U0∈G1∪G3 , then only the zeroϕ (U0, α ) = ϕ1(U0, α )is selected.

(ii) If U ∈G , then only the zerosϕ (U , α ) = ϕ (U , α )is selected.

Proof The second equation of (3.6) determines the u-value as u=u( ρ ) Taking the derivative with respect to ρ in the equation

α2

u( ρ ) ρ 2

= ( α0u0ρ0)2,

we get

α ( ρ ) α( ρ )(uρ )2+2α2(uρ ) 

u( ρ )ρ +u( ρ ) 

Thus, to prove the lemma, it is sufficient to show that the factor(u( ρ )ρ +u( ρ ))remains of a constant sign whenever( ρ ,u)

remains in the same domain Indeed, assume for simplicity that u0>0, then

u( ρ ) ρ +u( ρ ) = − κγργ−1

u +u

=u2− κγργ−1

which remains of a constant sign as long as( ρ ,u)remain in the same domain This completes the proof of Lemma 3.4 2

4 A well-balanced scheme based on stationary waves

In this section we will propose a numerical method to approximate the solution of the BN model (1.1)–(1.2) Our method

is relying on a given standard numerical scheme As seen earlier, the system is regrouped into three subsystems which have different behaviors For the first subsystem of the governing equations in the gas phase, the equilibrium states representing the effect of the source terms to the system will then be cooperated into the standard scheme The second subsystem has the conservative form and is treated in a usual way For the third subsystem of the compaction dynamics equation, we argue that the discretized equation in fact has more regularity than it is theoretically supposed due to the separation of waves and the properties of the solid contact Then, we invoke the Engquist–Osher scheme for this subsystem

Let us now present the details Given a uniform time step t, and a spacial mesh size x, setting x j=j x,j∈Z , and

t n=n t n∈N, we denote U n

j to be an approximation of the exact value U(x j, n) A CFL condition is also required on the mesh sizes:

λmax

U



|u g| + p

g( ρg), |u s| + p

s( ρs) 

<1, λ := t

To discretize the equations on the gas phase, or more precisely, the first subsystem (3.1), we use the following strategy which consists of two steps:

(i) First, we deal with the impact of the change of the volume fraction If the volume fraction changes, the nonconservative

term p g∂xαg is absorbed into stationary waves that produce equilibrium states Thus, these equilibrium states are obtained as the result of volume fraction change

(ii) Second, the equilibrium states obtained from the first step will move according to the governing equation where the volume fraction is constant This enables us to eliminate the volume fraction on both sides of the equations so that the subsystem becomes the usual isentropic gas dynamics

Thus, we set the dependent conservative variable

v=

ρg

We then take a suitable standard numerical flux for isentropic gas dynamic equations g(v,w) The first component of the well-balanced scheme for the first subsystem is defined by

v n+1

j =v n j− λ g

v n j,v n j+1,−



−g

v n j−1,+,v n j



The states v n j±, j∈Z , n∈N are defined as follows Set

v n j+1=



ρn

g , j+1

ρn

g , j+1u n g , j+1



We need to make the scheme always well-defined Therefore, it is necessary to define the approximate “relaxation” value for the volume fraction

αn g , ,Relaxj =max

αn g , j, αmin



αn g , j+1, ρn g , j+1,u n g , j+1



Proof Most of the proof was available in [26] However, for completeness, we will show the steps Assume