toro vazquez

7/28/2019 Toro Vazquez

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On Flux Vector Splitting for the Euler Equations:

Some Recent Results *

Eleuterio Toro

Laboratory of Applied MathematicsUniversity of Trento, [email protected]

http://www.ing.unitn.it/toro

* E F Toro and M E Vazquez-Cendon. Flux Splitting Schemes for the Euler Equations. Computers and Fluids.(Under review, April 2012).

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Table of contents

1 The Euler Equations and Flux Splitting

2 The Liou-Steffen Splitting

3 The Zha-Bilgen Splitting

4 A Flux-Splitting Framework

5 A Novel Splitting for the Euler Equations

6 Numerical Fluxes

7 Reinterpretation of Other Flux Splittings

8 Numerical Results for the Euler Equations

9 Other Potential Schemes for the Pressure System

10 Concluding Remarks

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The Euler Equations and Flux Splitting

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The Euler equations in one space dimension are

tQ+ xF(Q) = 0 , (1)

where Q is the vector of conserved variables and F(Q) is the flux vector,both given as

Q =

uE

, F(Q) =

u

u2 + pu(E+ p)

. (2)

Here is density, u is particle velocity, p is pressure and E is total energygiven as

E = (1

2u2 + e) . (3)

The specific internal energy e is, in general, a function of other variablesvia an equation of state. For example, e may be taken to be a function ofdensity and pressure, namely

e = e(, p) . (4)

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For ideal gases

e(, p) =p

( 1) , (5)

where 1 < < 3 is the ratio of specific heats. For air at moderatepressures and temperatures one uses = 1.4.

For solving numerically equations of the type (1) we adopt a conservativemethod of the form

Qn+1i = Qni tx [Fi+12 Fi 12 ] , (6)

where Fi+ 12

is the numerical flux. For background on the Euler equations

and conservative schemes of the form (6).

Flux vector splitting:

F(Q) = A(Q) + P(Q) , (7)

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The Liou-Steffen Splitting

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Liou and Steffen (1993) split the flux vector into and advection part A(Q)and a pressure part P(Q) as follows

F(Q) = A(Q) + P(Q) ,

with

A(Q) =

uu2

uH

, P(Q) =

0p0

, (8)

where

H =E+ p

(9)

is the enthalpy.

From the numerical point of view the aim is to obtain a numerical flux for(6) of the form

Fi+ 12

= Ai+ 12

+Pi+ 12

(10)

by finding partial advection and pressure fluxes Ai+12

and Pi+ 12

.

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To begin with, Liou and Steffen express A(Q) as

A(Q) = M

a

auaH

, (11)

where M = u/a is the Mach number and a =

p/ is the speed ofsound.

The advection flux is then taken as

Ai+ 12

= Mi+ 12

Ai+ 12

, (12)

with

Ai+ 12

=

a

au

aH

n

i

if Mi+ 12

0 ,

aauaH

n

i+1

if Mi+ 12

< 0 .

(13)

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The advection flux is upwinded according to advection speed implied inthe Mach number Mi+ 1

2

, which is split as

Mi+ 12

= M+i + Mi+1 , (14)

with

M =

14(M1)2 if | M | 1 ,12(M | M |) if | M | > 1 .

(15)

The pressure vector Pi+12

is constructed by splitting the pressure as

pi+ 12

= p+i + pi+1 , (16)

with two choices for the negative and positive components as follows

p = 1

2p(1

M) if

|M

| 1 ,

12p

(M|M|)M

if | M | > 1 , (17)

and

p =

14p(M1)2(2 M) if | M | 1 ,1

2

p (M|M|)

M

if|

M|

> 1 .(18)

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The Zha-Bilgen Splitting

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Zha and Bilgen (1993) split the flux vector into as in (7) with

A(Q) = u

u2uE

, P(Q) =

0p

pu

. (19)

Numerically, they propose fluxes Ai+ 12

and Pi+ 12

as follows.

Ai+12

= A+i +Ai+1 , (20)

whereAi = min(0, u

ni )Q

ni , A

+i = max(0, u

ni )Q

ni . (21)

For the pressure flux vector Zha and Bilgen use the splitting

Pi+ 12

= P+i +Pi+1 . (22)

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F h Zh d Bil d h Li S ff li i (16)
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For the p component Zha and Bilgen adopt the Liou-Steffen splitting (16),(17), while for the pu component they propose

(pu)i+ 12

= (pu)+i + (pu)i+1 , (23)

where

(pu)i = pni

uni if Mni 1 ,

12(u

ni ani ) if 1 < Mni < 1 ,

0 if Mni 1 ,

(24)

and

(pu)+i = pni

0 if Mni 1 ,12(u

ni + a

ni ) if

1 < Mni < 1 ,

uni if Mni 1 .

(25)

Finally the Zha-Bilgen numerical flux is

Fi+ 12

= A+i +P+i +A

i+1 +P

i+1 . (26)

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A Flux-Splitting Framework

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The framework

We propose to split system (1) via the flux splitting (7) into the twosystems

tQ+ xA(Q) = 0 ,

tQ+ xP(Q) = 0 ,

(27)

called respectively the advection system and the pressure system. The aimis then to compute a numerical flux as

Fi+12

= Ai+ 12

+Pi+12

, (28)

where Ai+12

and Pi+ 12

are obtained respectively from appropriate Cauchy

problems for the advection and pressure systems (27).

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Consider the Cauchy problem for the linear advection equation

tq(x, t) + xq(x, t) = 0 , < x < , t > 0 ,q(x, 0) = h(x) ,

(29)

where is a constant. The exact solution of IVP (29) after a time t is

q(x, t) = h(x t) . (30)

We now decompose the characteristic speed as

= + (1 ) = a + p , 0 1 , (31)

with definitions

a = , p = (1 ) , (32)so as to obtain two linear partial differential equations, namely

tq+ axq = 0 , tq+ pxq = 0 . (33)

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Now consider first the Cauchy problem for the advection equation
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Now consider first the Cauchy problem for the advection equation

tq+ axq = 0 ,q(x, 0) = h(x) ,

(34)

the solution of which after a time t1 is

q(x, t1) = h(x at1) . (35)Consider the Cauchy problem

tq+ pxq = 0 ,q(x, 0) = h(x at1) .

(36)

The exact solution of IVP (36), after a time t2, is

q(x, t2) = h(x at1 pt2) . (37)The combined solution of IVPs (34) and (36) for t1 = t2 = t is

q(x, t) = h(x

(a + p)t) = h(x

t) = q(x, t) . (38)

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The above result can be stated as the following proposition.

Proposition 3.1. The exact solution of the initial value problem (29) can

be obtained by solving in sequence the initial-value problems (34) and (36).

Remark 3.1. We note that in the wave decomposition (31),(32) of themodel problem (29) one can accept the characteristic speeds to bearbitrarily different. For example, for > 0, by taking a very small in

(31) we would have a


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Remark 3.3. For the full non-linear problem, such as the Euler equations,the proposed framework has two components: (a) the particular way thefull system is split into two subsystems, called here the advection andpressure systems, and (b) the numerical treatment of each subsystem toproduce corresponding advection and pressure numerical fluxes to make upthe numerical flux for the full system.

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A Novel Splitting for the Euler Equations

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Here we propose a new splitting for the Euler equations by noting that the
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Here we propose a new splitting for the Euler equations by noting that theflux may be decomposed thus

F(Q) =

uu2 + p

u(12u2 + e + p)

=

uu2

12u3

+

0p

u(e + p)

, (39)

with the corresponding advection and pressure fluxes defined as

A(Q) = u

u

12u2

, P(Q) =

0p

u(e + p)

. (40)

We note that the proposed advection flux A(Q) contains no pressureterms. All pressure terms from the flux F(Q), including that of the totalenergy E, are now included in the pressure flux P(Q). The advection flux

may be interpreted as representing advection of mass, momentum andkinetic energy. For the ideal gas case (5) the pressure flux (40) becomes

P(Q) =

0p

1pu

. (41)

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The advection system is

tQ+ xA(Q) = 0 , (42)

where Q = [,u,E]T and A(Q) as in (40) above. In quasi-linear formthe advection system becomes

tQ+M(Q)xQ = 0 , (43)

where M(Q) is the Jacobian matrix given as

M(Q) =

0 0 0

u2

2u 0

u3 32u2 0

. (44)

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It is easy to show that the eigenvalues of this matrix are

1 = 0 , 2 = 3 = u . (45)

There are only two linearly independent right eigenvectors, namely

R1 = 1

10

0

, R2 = 2

1u

12u2

. (46)

Thus the system is weakly hyperbolic, as there is no complete set oflinearly independent eigenvectors.

Regarding the nature of the characteristic fields, it is easy to show that the1-field is linearly degenerated and that the 2-field is genuinely non-linearif 2 = 0 and u = 0; otherwise it is linearly degenerate.

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The pressure system
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The pressure system

In terms of the conserved variables Q = [,u,E]T the pressure system is

tQ+ xP(Q) = 0 , (47)

with P(Q) as given in (40) above. In quasi-linear form the advectionsystem becomes

tQ+N(Q)xQ = 0 , (48)where N(Q) is the Jacobian matrix given as

N(Q) =

0 0 0

12( 1)u2 ( 1)u 1

u3 uE/ E/ 32u2 u

. (49)

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Th i l f N(Q) l l d i
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The eigenvalues ofN(Q) are always real and given as

1 =1

2u 1

2A , 2 = 0 , 3 =

1

2u +

1

2A , (50)

whereA =

u2 + 4a2 , a2 =

p

. (51)

Here a is the usual speed of sound for the full Euler equations.

In terms of physical variables the system reads

tV +B(V)xV = 0 , (52)

where

V =

u

p

, B =

0 0 00 0 1/

0 p u

. (53)

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Note that, since u < A =

u2 + 4a2, the system is always subsonic, that

is1 =

1

2u 1

2A < 0 < 3 =

1

2u +

1

2A . (54)

The right eigenvectors of matrix B in (53) corresponding to theeigenvalues (50) are

R1 =

02

(u A)

, R2 =

10

0

, R3 =

02

(u + A)

. (55)

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Numerical Fluxes
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Numerical Fluxes

In order to compute advection and pressure fluxes Ai+ 12

and Pi+ 12

we

consider the Riemann problem for each subsystem. We start with thepressure system.

To compute the flux for the pressure system we consider the Riemannproblem in terms of physical variables

tV +B(V)xV = 0 ,

V(x, 0) =

VL Vni if x < 0 ,VR

Vni+1 if x > 0 .

(56)

The solution of this problem has structure as shown in Fig. 1. The wavepattern is always subsonic, with a stationary contact discontinuity and twonon-linear waves to the left and right of the contact wave.

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t
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t

x

p u

(2 =1

2(u + A))

(1 =1

2(uA))

x = 0

Fig. 1. Structure of the solution of the Riemann problem for the pressuresystem.

u =CRuR CLuL

CR CL 2

CR CL (pR pL) ,

p =CRpL CLpR

CR CL+

1

2

CRCL

CR CL(uR

uL) ,

(57)

withCL = L(uL AL) ; CR = R(uR + AR) , (58)

where AL and AR are computed from (51).

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O ld i th li i ti b l i th t


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One could improve upon the linear approximation by applying the exactgeneralised Riemann invariants throughout. The result is the 2 2non-linear system for p and u

u

u2 + 4pL

= uL

u2L + 4pLL

,

u +

u2 + 4

pR

= uL +

u2R + 4

pRR

.

(59)

Finally, the numerical flux for the pressure system is given as

Pi+1

2 = p

0

11u

. (60)

Remark: no visible difference between exact and approximate

solutions.

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The advection system
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The advection system

Recall that in our splitting (40) the advection operator may be written thus

A(Q) = uu2

12u

3

= uK(Q) , K(Q) =

u12u

2

, (61)Advection ofK (mass, momentum and kinetic energy) with speed u. Herewe propose two algorithms.

Algorith 1 (TV scheme).

A(Q) = ui+1

2

K , (62)

where ui+ 1

2

is the intercell advection velocity taken as ui+ 1

2

from solution

(57) of the Riemann problem (56)

Ai+12

= ui+ 1

2

Kni if ui+ 1

2

> 0 ,

Kni+1 if ui+ 1

2

0 .(63)

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Algorith 2 (TV-AWS scheme). Here propose a weighted splitting scheme,
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g ( ) p p g p gwhich is a simple modification of the scheme proposed by Zha and Bilgen[?] for their advection system. The modified scheme is given as follows

Ai+12 = A+i +A

i+1 , (64)

with

Ai =1

2(1 )uniKni , A+i =

1

2(1 + )uniK

ni . (65)

Here = (uni ) =

uni + (uni )

2, (66)

with a small positive quantity, = 0.1, for example. The function (uni )allows a smooth transition from upwinding fully to the left and fully to theright, in the vicinity of uni = 0.

The resulting scheme from Algorithm 2, called the TV-AWS scheme, iseffectively a weighted averaged scheme and the Zha-Bilgen scheme isrecovered from it by simply setting the weight to be = sign(uni ).

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Summary of the present scheme
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Summary of the present scheme

In order to compute a numerical flux Fi+ 12

for the conservative scheme (6)

we proceed as follows:

Pressure flux. Evaluate the intercell pressure pi+ 1

2

and velocity ui+1

2

from the solution of the Riemann problem given in (57) to compute

the pressure flux Pi+1

2 as in (60). Advection flux. We have proposed two options. From algorithm 1

(TV scheme) we evaluate the advection flux Ai+ 12

as in (63).

Algorith 2 (TV-AWS) is described in equations (64) to (66).

Intercell flux. Compute the intercell flux Fi+12 as in (28), namelyFi+ 1

2

= Ai+ 12

+Pi+ 12

. (67)

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The Liou-Steffen scheme


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The Liou-Steffen splitting (1993) may be interpreted in our frameworkdefining the advection system as

tQ+ xA(Q) = 0 , (68)

with

A(Q) =

u

u

2

u(E+ p)

(69)

and the pressure system as

tQ+ xP(Q) = 0 , (70)

with

P(Q) =

0p

0

. (71)

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In terms of primitive variables V = [,u,p]T the pressure system can be


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p [, ,p] p yshown to hyperbolic with eigenvalues

1 = 2 = 0 , 3 =

(

1)u (72)

and three linearly independent eigenvectors

R1 = 1

10

0

, R2 = 2

01

0

, R3 = 3

01

( 1)u

. (73)

Here 1, 2 and 3 are scaling factors. Simple calculations show that thecharacteristic fields associated with 1 and 2 are linearly degenerate andthe characteristic field associated with 3 is genuinely non-linear.

Unfortunately we have not been able to find a straightforward pressurenumerical flux by solving the Riemann problem for this unusual hyperbolicsystem. Thus the re-interpretation of the Liou-Steffen splitting in ourframework has not been productive.

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The Zha-Bilgen splitting
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g p g

The Zha-Bilgen splitting (1993) assumes a very natural splitting that maybe interpreted in our framework as follows. The advection system is

tQ+ xA(Q) = 0 , A(Q) =

u

u2

uE

(74)

and the pressure system is

tQ

+ xP

(Q

) =0

,P

(Q

) =

0

ppu

. (75)

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In terms of primitive variables V = [,u,p]T the pressure system can beshown to be hyperbolic with real eigenvalues

1 = C , 2 = 0 , 3 = C , (76)

withC =

( 1)p/ , (77)

and three linearly independent right eigenvectors

R1 = 1

01

C

, R2 = 2

100

, R3 = 3

01

C

. (78)

Here 1, 2 and 3 are scaling factors.

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The Riemann problem for the Zha-Bilgen pressure system in terms of
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primitive variables is

tV + Z(V)xV = 0

V(x, 0) =

VL Vni if x < 0 ,

VR Vni+1 if x > 0 .

(79)

The structure of the solution is analogous to that shown in Fig. 1,

u =LCLuL + RCRuR

LCL + RCR (pR pL)

LCL + RCR,

p =R

CR

pL

+ L

CLpR

LCL + RCR L

CL

R

CR

LCL + RCR (uR uL) ,

(80)

withCL =

( 1)pL/L , CR =

( 1)pR/R . (81)

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However, application of the scheme to cell i for one time step gives


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En+1i =p

1 + ip , i =1

2

t

x

p(

1R

1R

) . (83)

Application of the scheme to cell i + 1 gives an analogous expression butwith i+1 = i. In order to preserve the contact discontinuity unalteredone requires i = 0, which is not satisfied by the Zha-Bilgen scheme, asseen in (83).

Distance

Density

0 0.25 0.5 0.75 10.7

0.9

1.1

1.3

1.5

ZB-orig

Exact

Fig. 2. Test 6: Stationary isolated contact. Exact (line) and numericalsolution (symbols) using the Zha-Bilgen original scheme (ZB-orig).

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Proposition 4.2. The Toro-Vazquez splitting (TV) along with theirnumerical method can recognise exactly isolated stationary contactdiscontinuities for the Euler equations.

Proof. Define the problem for a stationary, isolated contact discontinuityfor the ideal gas Euler equations with initial condition as in (82). Assumethe discretization of the domain [xL, xR] is such that the cell just to theleft of the discontinuity is i and that immediately to the right of the

discontinuity is i + 1. Application of the TV scheme to any cell away fromcells i and i + 1 leaves the flow indisturbed. Let us now apply the schemeto cell i for one time step. First we need the solution (57) of the Riemannproblem with initial data (82). Clearly u = ui+ 1

2

= 0 and p = pi+ 12

= p.

Then it is easy to verify that the state Qn+1

i = Qn

i and thus the isolatedstationary contact is preserved exactly. Application of the scheme to celli + 1 gives an analogous result and the proposition is thus proved.

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Proposition 4.3. The Zha-Bilgen splitting along with the Godunov-type
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numerical method of section 4.2 can recognise exactly isolated stationarycontact discontinuities for the Euler equations.

Proof. The proof is straightforward and is thus omitted.

Distance

Density

0 0.25 0.5 0.75 10.9

1

1.1

1.2

1.3

1.4

1.5

TV

Exact

Distance

Density

0 0.25 0.5 0.75 10.7

0.9

1.1

1.3

1.5

ZB-God

Exact

Fig. 3. Test 6: Stationary isolated contact. Exact (line) and numericalsolutions (symbols).

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Numerical Results for the Euler Equations

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1

TV

Exact

1

TV AWS


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Frame 002 23 Jul 2011

Distance

Densitty

0 0.25 0.5 0.75 10.1

0.4

0.7

Exact

Distance

Dens

ity

0 0.25 0.5 0.75 10

0.2

0.4

0.6

0.8

TV-AWS

Exact

Test 1 (sonic flow). Exact (line) and numerical solutions (symbols) usingtwo numerical schemes (TV and TV-AWS) for the flux splitting of this

paper.

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1

LS

1


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Distance

Dens

ity

0 0.25 0.5 0.75 10.1

0.4

0.7

Exact

Distance

Dens

ity

0 0.25 0.5 0.75 10.1

0.4

0.7

ZB-orig

Exact

Test 1 (sonic flow). Exact (line) and numerical solutions (symbols) usingtwo numerical schemes: Liou-Steffen (LS) and Zha-Bilgen (ZB-orig).

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1

TV

Exact

1

Exact
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Distance

Dens

ity

0 0.25 0.5 0.75 1

0

0.2

0.4

0.6

0.8

Exact

Distance

Dens

ity

0 0.25 0.5 0.75 1

0

0.2

0.4

0.6

0.8

TV-AWS

Test 2 (low density). Exact (line) and numerical solutions (symbols) usingtwo numerical schemes (TV and TV-AWS) for the flux splitting of this

paper.

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6

TV

E act

6

TV-AWS
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Distance

Dens

ity

0 0.25 0.5 0.75 10

1

2

3

4

5

Exact

Distance

Dens

ity

0 0.25 0.5 0.75 10

1

2

3

4

5Exact

Test 3 (very strong shock). Exact (line) and numerical solutions (symbols)using two numerical schemes (TV and TV-AWS) for the flux splitting of

this paper.

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35

TV

35

TV-AWS

E
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Distance

Dens

ity

0 0.25 0.5 0.75 15

10

15

20

25

30Exact

Distance

V2

0 0.25 0.5 0.75 15

10

15

20

25

30Exact

Test 4 (collision of two strong shocks). Exact (line) and numericalsolutions (symbols) using two numerical schemes (TV and TV-AWS) for

the flux splitting of this paper.

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6

7

TV

Exact6

7

TV-AWS

Exact
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Distance

Dens

ity

0 0.25 0.5 0.75 10

1

2

3

4

5

6

Distance

Dens

ity

0 0.25 0.5 0.75 10

1

2

3

4

5

6 Exact

Test 5 (non-isolated stationary contact discontinuity). Exact (line) andnumerical solutions (symbols) using two numerical schemes (TV and

TV-AWS) for the flux splitting of this paper.

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1.5 1.5

TV-AWS


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Distance

Dens

ity

0 0.25 0.5 0.75 10.9

1

1.1

1.2

1.3

1.4

TV

Exact

Distance

Dens

ity

0 0.25 0.5 0.75 10.9

1

1.1

1.2

1.3

1.4 Exact

Test 6 (Isolated stationary contact discontinuity). Exact (line) andnumerical solutions (symbols) using two numerical schemes (TV and

TV-AWS) for the flux splitting of this paper.

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1.5 1.5
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Distance

Dens

ity

0 0.25 0.5 0.75 10.7

0.9

1.1

1.3

ZB-orig

Exact

Distance

Dens

ity

0 0.25 0.5 0.75 10.7

0.9

1.1

1.3

ZB-God

Exact

Test 6 (Isolated stationary contact discontinuity). Exact (line) andnumerical solutions (symbols) using two numerical schemes: the

Zha-Bilgen original scheme (ZB-orig) and the Zha-Bilgen splitting with

present Godunov-type numerical approach (ZB-God).

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6

7

WAF

God+Ex. R S

TV

6

7

WAF

God+Ex. R S

TV-AWS
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Distance

Dens

ity

0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6 TV

Distance

Dens

ity

0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6 TV-AWS

Test 7 (Woodward and Colella blast wave problem). Reference solutions(WAF and Godunovs method with exact Riemann solver) and numerical

solutions from two numerical schemes of this paper: TV (top) and

TV-AWS (bottom).

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Other Potential Schemes for the Pressure System

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Concluding Remarks

New flux splitting New way of dealing with pressure term Scheme captures contact discontinuity as well as AUSM Our scheme is more robust and more accurate than AUSM Other schemes for pressure system under study

Best combination: explicit for advection implicit for pressure

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toro vazquez

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