DSpace at VNU: Building fast well-balanced two-stage numerical schemes for a model of two-phase flows

The volume fractions are constraint by the relationIn our recent work[40], a Roe-type scheme was constructed by using the admissible solid contacts at each node to absorbthe nonconservat

Building fast well-balanced two-stage numerical schemes

for a model of two-phase flows

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

by using a convex combination of the numerical flux of a stable Lax–Friedrichs-typescheme and the one of a higher-order Richtmyer-type scheme Numerical schemes con-structed in such a way are expected to get the interesting property: they are fast and stable.Tests show that the method works out until the parameter takes on the value CFL, and soany value of the parameter between zero and this value is expected to work as well All theschemes in this family are shown to capture stationary waves and preserves the positivity

of the volume fractions The special values of the parameter 0; 1=2; 1=ð1 þ CFLÞ, and CFL inthis family define the Lax–Friedrichs-type, FAST1, FAST2, and FAST3 schemes, respectively.These schemes are shown to give a desirable accuracy The errors and the CPU time of theseschemes and the Roe-type scheme are calculated and compared The constructed schemesare shown to be well-balanced and faster than the Roe-type scheme

Ó 2013 Elsevier B.V All rights reserved

E-mail address: mdthanh@hcmiu.edu.vn

Contents lists available atScienceDirect

Commun Nonlinear Sci Numer Simulat

j o u r n a l h o m e p a g e : w w w e l s e v i e r c o m / l o c a t e / c n s n s

Throughout, we use the subscripts g and s to indicate the quantities in the g-phase (referred to as the gas phase), and in thesolid phase (referred to as the solid phase), respectively However, our study in this work can be applied for more general mate-rials For example, one of the two phases or both may be liquid The notationsak; qk; uk; pk; k ¼ g; s, stand for the volume frac-tion, density, velocity, and pressure in the k-phase, k ¼ g; s, respectively The volume fractions are constraint by the relation

In our recent work[40], a Roe-type scheme was constructed by using the admissible solid contacts at each node to absorbthe nonconservative terms The states on both sides of these solid contacts are incorporated into a Roe-type matrix of thedecoupling system of(1.1)which is obtained from(1.1)by letting the volume fractions be constant A similar process couldalso be made by using another numerical flux, for example the one of the Lax–Friedrichs scheme to form a Lax–Friedrichs-type scheme However, this Lax–Friedrichs-type scheme has less accuracy then the Roe-type scheme, as seen in Section4.This is probably because the Lax–Friedrichs scheme is, though stable, too diffusive Furthermore, the incorporation of thestates on both sides of the solid contacts mentioned above into a higher-order scheme such as the Lax–Wendroff or Richm-yer’s scheme does not yield satisfactory results: the scheme is numerically unstable, where large oscillations appear shortly

in numerical tests Customarily, a well-balanced scheme is the one which can preserve the steady state solution exactly.Motivated by the above argument, we aim to build in this paper a set of numerically stable schemes that can be faster andhave a better accuracy than the Roe-type scheme For this purpose, we form a one-parameter family of numerical fluxes byusing convex combinations of the numerical fluxes of the Lax–Friedrichs scheme and the second-order Richtmyer’s with aparameter h 2 ½0; 1 The states Un

j1;on the other side of the solid contacts at xj1=2from any given states Un

j1are then porated in numerical fluxes in this family to produce new schemes The values h ¼ 0; 1=2; h ¼ 1=ð1 þ CFLÞ and CFL in thisfamily define a Lax–Friedrichs-type, FAST1, FAST2, and FAST3 schemes, respectively Thus, the FAST1 and FAST2 schemesare formed in a similar way as the FORCE and GFORCE schemes Recall that the FORCE and GFORCE schemes are convex com-binations of the Lax–Friedrichs scheme and Lax–Wendroff scheme, see[29,30,9] Tests of Lax–Friedrichs-type, FAST1, FAST2,and FAST3 schemes are presented Errors, order of convergence, numbers of iterations, CPU time are evaluated.All the testsshow desirable approximations to the exact solutions The results are compared with a newly constructed Roe-type scheme

incor-[40] Tests show that the FAST3 scheme gives the better results than the Roe-type scheme Naturally, the same result is pected for schemes corresponding to the values of the parameter h closed to CFL, at least Moreover, we show that ourschemes are well-balanced Observe that the restrictive attention to the isentropic case may cause a certain limitation forapplications, and would motivate for future developments for the general case

ex-We note that numerical approximations of nonconservative systems have attracted attention of many authors Numericalwell-balanced schemes for a single conservation law with a source term are presented in[15,16,5,6,3] Numerical schemesfor multi-phase multi-pressure models were presented in[21,25,1,26,13,39,36] Various numerical schemes for two-fluidmodels of two-phase flows were constructed in[42,38,35] Well-balanced schemes for other nonconservative hyperbolicsystems were built in [20,19,24,3,37,8,17].The Riemann problem for various nonconservative hyperbolic systems wasstudied in[22,33,23,14,32,2] Shock waves in two-fluid models of two-phase flows were studied in[18,34] Some recentGodunov-type schemes for various fluid flow models are presented in[31,27,24,28] See also the references therein.The organization of this paper is as follows In Section2 we present basic concepts of the system(1.1): non-stricthyperbolicity, discontinuities, and admissible solid contact waves In Section3 we construct a one-parameter family ofwell-balanced schemes First, we describe a family of numerical fluxes by using convex combinations of the numerical fluxes

of a stable Lax–Friedrichs and a higher-order Richmyer’s schemes Then, we show how to incorporate admissible solidcontacts into these numerical fluxes to obtain a numerical scheme Section4is devoted to numerical tests, where testsfor the well-balanced Lax–Friedrichs-type scheme, FAST1, FAST2, FAST3, and Roe-type schemes are carried out Finally, inSection5we draw several conclusions and discussions

1CCCC

ð2:2Þ

and the ‘‘enthalpy in the isentropic case’’ hkis defined by

2.3 Admissible solid contacts

Let U0¼ ðqg0;ug0;qs0;us0;ag0Þtbe a given and fixed state We look for any state U ¼ ðqg;ug;qs;us;agÞtthat can be nected with U0by a discontinuity where us constant, as seen by(2.10) In this case, the discontinuity was shown to beassociated with the 5th characteristic field, and so it is referred to as a 5-contact wave, or a solid contact, see[11,2,39,32].Moreover, the state U satisfies the equations

ag;qgand ugcan be found using(2.11) The solid pressure is given by(2.12), and therefore the solid density can be calculated

by using an equation of state of the formqs¼qsðpsÞ Thus, the state corresponding state U ¼ ðqg;ug;qs;us;agÞton the otherside of the solid contact can be completely determined by(2.11)and(2.12) If we choose the left-hand state to be U¼ U0,then the corresponding right-hand state is given by Uþ¼ U satisfying(2.11)and(2.12); if we choose the right-hand state to

be Uþ¼ U0, then the corresponding left-hand state is given by U¼ U satisfying(2.11)and(2.12) These states Uwill mine the points

on both sides of the solid contact, which will be used in the construction of well-balanced schemes in the next section.For simplicity, in the rest of this subsection we will drop the subindex ‘‘g’’ for the quantities in the gas phase Moreover,

we assume that the fluid in the gas phase is isentropic and ideal, that is, the equation of state on the gas phase is of the form

(2.3) Then, it follows from(2.11)that the gas density satisfies the following nonlinear algebraic equation

The functionWin(2.14)is strictly convex It possesses two zeros whenever

aPaminðU0Þ :¼ a0q0ju0 usj

ffiffiffiffiffiffijc

lð c þ1Þðu0 usÞ2þ 2

c þ1qc 1 0

fol-(MC) Any contact wave does not cross the resonant surface C

It is not difficult to verify that

WðU0;a;qÞ > 0; for q6q:¼ a0ðu0 usÞq0

!1=ð c 1Þ

:

ð2:16Þ

The admissible zero can be determined and calculated as follows

(i) For U02 G1[ G3, the admissible value of the density is the smaller zeroq¼u1ðU0;aÞ ofW, which can be computedusing Newton’s method starting atqdefined by(2.16);

(ii) For U02 G2, the admissible value of the density is the larger zeroq¼u2ðU0;aÞ ofW, which can be computed usingNewton’s method starting atqdefined by(2.16)

3 A set of well-balanced schemes

3.1 The underlying numerical fluxes for the decoupling system

It is derived from(2.10)that if ½as ¼ 0, then the volume fractions remain constant across the discontinuity The governingequations of the system(1.1)is therefore reduced to the following decoupling system of two independent sets of isentropicgas dynamics equations in both phases

1CC

Lax–Friedrichs scheme for the decoupling system

The numerical flux of the Lax–Friedrichs scheme is given by

gLFðU; VÞ ¼1

2ðf ðUÞ þ f ðVÞÞ 

1

Richtmyer’s scheme for the decoupling system

Richtmyer’s scheme is a second-order scheme that has the numerical flux

Convex combinations the numerical fluxes of the Lax–Friedrichs scheme and Richtmyer’s scheme

These schemes are constructed by using convex combinations of the Lax–Friedrichs scheme and Richtmyer’s scheme cisely, let us be given the Lax–Friedrichs numerical flux gLFðU; VÞ and the Richtmyer numerical flux gRðU; VÞ We define

where gLFand gRare given by(3.6) and (3.7), respectively

A Roe matrix for the decoupling system

and the notations hold for the corresponding quantities in both phases A Roe scheme for the decoupling system(3.1)is ven by

1CC

The corresponding right-eigenvectors can be chosen as

0BBB

1CC

C; r2ðUL;URÞ ¼

1



k2ðUL;URÞ00

0BBB

1CCC

and



r3ðUL;URÞ ¼

001



k3ðUL;URÞ

0BBB

1CC

C; r4ðUL;URÞ ¼

001



k4ðUL;URÞ

0BBB

1CC

C:

The corresponding left-eigenvectors of the Roe matrix can be chosen as

Define the coefficients

aiðUL;URÞ ¼ liðUL;URÞðUR ULÞ; i ¼ 1; 2; 3; 4:

A Roe numerical flux for the system(3.1)is given by

gRoeðV; WÞ ¼1

2ðFðVÞ þ FðWÞÞ 

12

X4 i¼1

where V; FðVÞ are given by(3.2)

3.2 Building two-stage well-balanced schemes

It follows from the equation of conservation of mass in the solid phase and the compaction dynamics equation that

g;jþ1an g;jÞ

jþ1to an admissible state denoted by Vnjþ1; withthe gas volume fractionan

j; The volume fraction change across the node x ¼ xj1=2¼ ðj  1=2ÞDx; t ¼ tn¼ nDt, creates a 5-contact discontinuityjumping from the given state Vnj1with the gas volume fractionan

j1to an admissible state denoted by Vnj1;þwith thegas volume fractionan

j.The corresponding states Vnj1;are of the form(2.13), and can be computed using the strategy described in the Section

2.3 Moreover, to make sure that the process of computing the admissible solid contacts always works, we suggest an tional computing technique For example, if the gas phase is ideal, we can substitute the volume fractionan

addi-g;jby its ‘‘relaxed’’value defined by

an;Relax

g;j;aminðan

g;j1;qn g;j1;un g;j1Þ

when computing Vnj1;, respectively, whereaminis defined by(2.15)

A one-parameter family of numerical schemes for(1.1)is constructed as follows The state Vn

jþ1;is substituted for Vn

jþ1,and the state Vn

j1;þis substituted for Vn

j1into the scheme(3.4), respectively, j 2 Z; n ¼ 0; 1; 2; , where the numerical flux

g ¼ ghis defined by(3.8) Together with(3.12), this yields a one-parameter family of schemes defined by

anþ1

g;j ¼an

g;jDt

Dx uþ;n s;j ðan g;jan g;j1Þ þ u;n s;j ðan g;jþ1an g;jÞ

Taking h ¼ 0 in(3.14), one obtains a Lax–Friedrichs-type (LF-type) scheme We define the FAST1, FAST2and FAST3 schemes to bethe schemes defined by(3.14)by taking h ¼ 0; 1=2; 1=ð1 þ CFLÞ, and CFL, respectively.

A Roe-type scheme was constructed in[40]by

anþ1

g;j ¼an

g;jDt

Dx uþ;n s;j ðan g;jan g;j1Þ þ u;ns;j ðan

g;jþ1an g;jÞ

where gRoeis given by(3.10)

The schemes(3.14)possesses nice properties as in the following theorem

Proposition 3.1 The following conclusions hold

(i) (Well-balanced scheme)The schemes(3.14)capture exactly steady state solutions This means that for any stationary tact wave, it holds that

con-Unþ1

where U ¼ ðqg;ug;qs;us;agÞ Thus, the schemes(3.14)are well-balanced

(ii) (Positivity of volume fractions) The well-balanced schemes(3.14)preserve the positivity of the volume fractions Thismeans that ifa0

k;j>0 for all j 2 Z, thenan

qn

g;j1;þ¼qn

g;j; un g;j1;þ¼ ung;j;

It therefore follows from(3.14)that

Vnþ1

From(3.17) and (3.18)we obtain(3.16)

The proof of (ii) is omitted, since it is similar to the one of Theorem 4.1 in[36].Proposition 3.1is completely proved h

4 Numerical tests

This section is devoted to numerical tests, where approximate solutions obtained by the schemes(3.14)are comparedwith the exact solution of the Riemann problem for(1.1) For these tests, both phases are assumed to have an equation ofstate of the form(2.3) The exact solution of the Riemann problem consists of elementary waves: shocks, rarefaction waves,

Fig 3 Test 2 Approximate solution by the two-stage method using different underlying numerical fluxes: Lax–Friedrichs, Roe, and FAST3 with 250 mesh

and a solid contact As seen above, the volume fractions change only across the solid contact, and the shocks and rarefactionwaves behave as in the usual gas dynamics equations of each individual phase Therefore, we will make five numerical testsrelying on the location and the sign of the solid velocity of the solid contact in the exact Riemann solution The parametersfor Tests 1–4 are

jg¼ 1; js¼ 0:4; cg¼ 1:4; cs¼ 1:6;

while the parameters for dioxygen and water at the temperature 100 °C are taken for Test 5

4.1 Numerical test 1: stationary contact discontinuities

This test will verify the well-balanced property of the proposed family of schemes The approximate solution will be puted at the time t ¼ 0:1 on the interval ½1; 1 of the x-space with 1000 mesh points We take the underlying numerical flux

com-to be the one of the LF-type scheme with CFL = 3/4 Consider the Riemann problem for(1.1)and (1.2) with the initial datagiven inTable 1

It is not difficult to check that in this case the Riemann solution is a stationary 5th contact wave TheFig 2shows that thestationary contact wave is well captured

4.2 Numerical test 2: solid contact in supersonic region with zero velocity

In this test, we consider the Riemann problem for(1.1)and (1.2) with the initial data given inTable 2 The volume fractionjump is large: jaLaRj ¼ 0:5 The Riemann solution has a stationary contact discontinuity The exact Riemann solution be-gins with a 1-shock wave from U to U This shock is followed by a 2-rarefaction wave from U to U , followed by a 3-shock

wave from U2to U3, followed by a solid contact from U3to U4and then followed by a 4-shock wave from U4to UR Thesestates are given inTable 3 The velocity of the solid contact between U and U is equal to zero It is not difficult to check

Fig 4 Test 2 Approximate solutionusing the underlying FAST3 numerical flux The approximate solution is computed at the time t ¼ 0:1 on the interval

½1; 1 with 250, 1000, and 4000 mesh points, and is compared with the exact solution.

Table 5 Test 3 – The initial values.