On Flux Vector Splitting for the Euler Equation

8/14/2019 On Flux Vector Splitting for the Euler Equation

1/58

On Flux Vector Splitting for the Eluer Equations

Eleuterio ToroLaboratory of Applied Mathmatics

University of Trento, Italy

[email protected]

http://www.ing.unitn.it/toro
mailto:[email protected]:[email protected]


2/58

Table of Contents

1 The Euler equation 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 Application and Extension of the scheme11 Concluding Remarks


3/58

Why another flux vector splitting scheme?

Intermediate characteristic fields have the answer.

Classical FVS schemes (Warming-Beam, van Leer, Zha-

Bilgen) are unsuitable . Excessive Numerical diffusion.

Liou-Steffen scheme (AUSM) is an exception but

We present a new framework for construct FVS schemes.Clear Basis

The Zha-Bilgen splitting can be recovered using the new

frame

We present a new splitting (Toro-Vazquez) for the Euler

Equations


4/58

Test 6Stationary isolated contact. Exact (line) and numerical solutions

(symbols)


5/58


6/58

1/2

1/2

1/2

1

if 0

if 0

n

i

i

i n

i

i

a

au M

aHA

a

au M

aH


7/58

The advection flux is upwind according to advection speed implied in

the Mach number +/, which is split as

with

The pressure vector +/is constructed by splitting the pressure as(16)

with two choices for the negative and positive components as follows

and (17)

1/2 1i i iM M M

211 if 1

4

1 if 1

2

M M

M

M M M

1/2 1i i ip p p

11 if 1

2

1 if 1

2

p M M

pM M

p MM

211 2 if 1

4

1 if 1

2

p M M M

pM M

p MM


8/58

The Zha-Bilgen Spliting


9/58

Zha and Bilgen (1993) split the flux vector into

with

= , = 0 .

Numerically, they propose fluxes

+/

and

+/as follows.

where

For he pressure flux vector Zha and Bilgen use the

F Q A Q P Q

1/2 1i i iA A A

min 0, , max 0, .n n n ni i i i i iA u Q A u Q

1/2 1i i iP P P


10/58

For the p component Zha and Bilgen adopt the Liou-Steffen Splitting

(16), (17), while for the pu compenent they propose

where

,

Finally the Zha-Bilgen numerical flux is

1/2 1i i ipu pu pu

if 1

1

if -1< 12

0 if 1

n n

i i

n n n n

i i i ii

n

i

u M

pu p u a M

M

0 if 1

1

if -1< 12

if 1

n

i

n n n n

i i i ii

n n

i i

M

pu p u a M

u M

1/2 1 1i i i i iF A P A P


11/58

A Flux-Splitting Framework


12/58

The Framework

We propose to split system

= 0via

= into two systems = 0 ,

= 0 ,, (27)

called respectively the advection system and the pressure system.

The aim is then to compute the numerical flux as+/= +/ +/,where +/and +/are obtained respectively fromappropriate Cauchy problems for the advection and pressure

systems (27).


13/58

Consider the Cauchy problem for the linear advection equation

, , = 0, < < , > 0, , = (29)where is a constant. The exact solution of IVP (29) after a time is , = .We now decompose the characteristic speed

as

= 1 = , 0 1, (31)with definitions = , = 1 , (32)so as to obtain two linear partial differential equations, namely

= 0, = 0.


14/58

Now consider the first Cauchy problem for the advection equation

= 0, ,0 = , (34)the solution of which after a time is , = .Consider the Cauchy problem

= 0, , 0 = ,. (36)The exact solution of IVP (36), after a time , is

, = .

The combined solution of IVPs (34) and (36) for = = is , = = = , .


15/58

The above result can be started 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 the

model problem (29) one can accept the characteristic speeds to be

arbitrarily different. For example, for

> 0, by taking a very small

in (31) we would have , situation that resembles the slowadvection waves and the fast pressure waves.

Remark 3.2 From the numerical point of view, Proposition 3.1

suggests a way to compute a numerical flux for IVP (29) by

computing numerical fluxes for IVPs (34) and (36). This would leadto split flux vector splitting methods and could potentially be of use to

deal with systems in which there is large disparity in the magnitude of

the wave speeds present. This line of enquiry has not been pursued, to

my knowledge.


16/58

A novel splitting for Euler equations


17/58

Here we propose a new splitting for the Euler equations by noting that the

flux may be decomposed thus

=

=

1/2 0 ,with the corresponding advection and pressure flux defined as

=

1/2 , = 0

, (40)We note that the proposed advection flux A contains no pressure terms.All pressure terms from the flux , including that of the total energy ,are now included in the pressure

. The advection flux may be

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

= 0

/ 1

.


18/58

The advection system is = 0,where = , , and A as in (40) above. In quasi-linear formthe advection system becomes

= 0,where M is the Jacobian matrix given as = 0 0 0 2 0

3/2 0.


19/58

It is easy to show that the eigenvalues of this matrix are= 0, = = .There are only two linearly independent right eigenvectors, namely

= 100 , =

11/2

.

Thus the system is weakly hyperbolic, as there is no complete set of

linearly independent eigenvectors.

Regarding the nature of the characteristic fields, it is easy to show that

the -filed is linearly degenerated and that the -filed is genuinelynon-linear if 0and 0; otherwise it is linearly degenerate.


20/58

The Pressure System

In terms of the conserved variables

= , , the pressure

system is = 0,with as given in (40) above. In quasi-linear form the

pressure system becomes

= 0,where is the Jacobian matrix given as

=

0 0 01/2 1 1 1 / 3/2

.


21/58

The eigenvalues of are always real and given as= , = 0, = .(50)where = 4, = /.(51)Here

is the usual speed of sound for the full Euler equations. In

terms of physical variables the system reads = 0,where

= , =

0 0 00 0 1 / 0 .(53)


22/58

Note that since < = 4, the system is always subsonic,that is= < 0 < = 0 < = .

The right eigenvectors of

in (53) corresponding to the

eigenvalues (50) are

= 02 , =

100 , = 02 .


23/58

Numerical Fluxes

In order to compute advection and pressure fluxes

+/and

+/we consider the Riemann problem for each subsystem. Westart with the pressure system.

To compute the flux for the pressure system we consider the

Riemann problem in terms of physical variables

The solution of this problem has structure as shown in Fig. 1. Thewave pattern is always subsonic, with a stationary contact

discontinuity and two non-linear waves to the left and right of the

contact wave.

L

R 1

0,

if 0 ,,0

if 0 .

t x

n

i

n

i

V B V V

V V xV x

V V x

(56)


24/58

with

whereandare computed from (51).

Fig. 1. Structure of the solution of the Riemann problem for the pressure system

R R L L* R L

R L R L

R L L R R L

* R L

R L R L

2 ,

1.

2

C u C uu p pC C C C

C p C p C C p u u

C C C C

L L R R ; ,L L R RC u A C u A

(57)


25/58

The Advection System

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

written thus

=

1/2 = , =

1/2 ,

Advection of (mass, momentum and kinetic energy) with speedu. Here we propose two algorithms.Algorithm 1 (TV scheme)

Where is the intercell advection velocity taken as fromsolution (57) of the Riemann problem (56)

* 1/2 ,iA Q u K

*

1/2*

1/2 1/2 *

1 1/2

if 0 ,

if 0 .

n

i i

i i n

i i

K uA u

K u

*

1/2iu *

1/2iu

(62)

(61)

(63)


26/58

Algorithm 2 (TV-AWS scheme). Here propose a weighted splitting

scheme, which is a simple modification of the scheme proposed by

Zha and Bilgen (1963) for their advection system. The modified

scheme is given as follows

with

Here

with a small positive quantity, , for example. The function

allows a smooth transition from upwinding fully to the left and fully to

the right, in the vicinity of .The resulting scheme from Algorithm 2, called the TV-AWS scheme,

is effectively a weighted averaged scheme and the Zha-Bilgen scheme

is recovered from it by simply setting the weight to be

1/2 1i i iA A A

1/ 2 1 , 1/ 2 1 .n n n ni i i i i iA w u K A w u K

2,

nn ii

n

i

uu

u

=0.1 niu

0niu

.nisign u

(64)

(65)

(66)


27/58

Summary of the present scheme

In order to compute a numerical flux

+/for the conservative

scheme (6) we proceed as follows:

Present flux. Evaluate the intercell pressure and velocity

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

compute the present flux as in (60).

Advection. We have proposed two options. From algorithm 1(TV scheme) we evaluate the advection flux as in (63).

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

Intercell flux. Compute the intercell flux as in (28), namely

*

1/2ip

1/2iP

1/2 1/2 1/2i i iF A P

*

1/2iu

1/2iA

1/2iF

(67)


28/58

Reinterpretation of other flux splitting


29/58

The Liou-Steffen scheme

The Liou-Steffen splitting (1993) may be interpreted in our

framework defining the advection system as = 0,with

=

.and the pressure system as = 0,with

= 00 .

(68)

(71)

(69)

(70)


30/58

In terms of primitive variables = , , the pressure systemcan be shown to hyperbolic with eigenvalues

= = 0, = 1 . (72)and three linearly independent eigenvectors=

100

, = 010

, = 01

1 .(73)

Here ,and are scaling factors. Simple calculations showthat the characteristic fields associated with and are linearlydegenerate and the characteristic field associated with isgenuinely non-linear.

Unfortunately we have not been able to find a straightforwardpressure numerical flux by solving the Riemann problem for this

unusual hyperbolic system . Thus the re-interpretation of the

Liou-Steffen splitting in our framework has not been productive.


31/58

The Zha-Bilgen splitting

The Zha-Bilgen splitting (1993) assumes a very natural splitting

that may be interpreted in our framework as follows. The

advection system is = 0,with

= .and the pressure system as

= 0,

with

= 0 .(75)

(74)


32/58

In terms of primitive variables = , , the pressure systemcan be shown to hyperbolic with eigenvalues= , = 0, = , (76)with

C = 1 / , (77)and three linearly independent right eigenvectors=

01

, = 100

, = 01

.(78)

Here ,and are scaling factors.


33/58

The Riemann problem for the Zha-Bilgen pressure system in terms

of primitive variables is

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

L

R 1

0,

if 0 ,,0

if 0 .

t x

n

i

n

i

V Z V V

V V xV x

V V x

(79)

with

L L L R R R R L

*

L L R R L L R R

R R L L L R L L R R

* R L

L L R R L L R R

,

.

C u C u p pu

C C C C

C p C p C C p u u

C C C C

L L L R R R 1 / ; 1 / .C p C p

(80)

(81)


34/58

Contact Discontinuity

Proposition 4.1. The Zha-Bilgen splitting along with the Zha-

Bilgen numerical scheme cannot sustain isolated stationary contact

discontinuities for the Euler equations.

Proof. Define the problem for a stationary, isolated contact

discontinuity for the ideal gas Euler equations with initial

condition

with < < . Assume the discretization of , such thatthe contact discontinuity is between cells and 1. Applicationof the Zha-Bilgen scheme to any cell away from cells and 1leaves the flow undisturbed.

(82)

L R

L 0

R 0

,0 0, , 0 : constant, such that ,

if ,,0

if ,

u x p x p x x x x

x xx

x x


35/58

However application of the scheme to cell for one time stepgives

Application of the scheme to cell 1gives an analogousexpression but with + = . In order to preserve the contactdiscontinuity unaltered one requires = 0, which is not satisfiedby the Zha-Bilgen scheme, as seen in (83).

1

R R

1 1 1 , .

1 2

n

i i i

p tE p p

x

(83)

Fig. 2. Test 6: Stationary isolated contact. Exact (line) and numerical

solution (symbols) using the Zha-Bilgen original scheme (ZB-orig)


36/58

Proposition 4.2. The Toro-Vazquez splitting (TV) along with their

numerical method can recognise exactly isolated stationary contact

discontinuities for the Euler equations.

Proof. Define the problem for a stationary, isolated contactdiscontinuity for the ideal gas Euler equations with initial

condition as (82). Assume the discretization of the domain , such that the cell just to the left of the discontinuity is and thatimmediately to the right of the discontinuity is 1. Applicationof the TV scheme to any cell away from cells and 1leavesthe flow undisturbed. Let us now apply the scheme to cell forone time step. First we need the solution (57) of the Riemann

problem with initial data (82). Clearly

= +/= 0and

= +/= . Then it is easy to verify that the stateand thus the isolated stationary contact is preserved exactly.

Application of the scheme to cell 1 gives analogous result andthe proposition is thus proved.

1n n

i iQ Q


37/58

Proposition 4.3. The Zha-Bilgen splitting along with the

Godunov-type numerical method of section 4.2 can recognize

exactly isolated stationary contact discontinuities for the Euler

equations.Proof. The proof is straightforward and is thus omitted.

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

solution (symbols) .


38/58

Numerical Results for the Euler Equations


39/58

Two classes of test problems

Test of Woodward and Colella (1984). Reference solution:WAF.

(84)

Test

1 1.0 0.75 1.0 0.125 0.0 0.1

2 1.0 -2.0 0.4 1.0 2.0 0.4

3 1.0 0.0 1000.0 1.0 0.0 0.01

4 5.99924 19.5975 460.894 5.99242 -6.19633 46.0950

5 1.0 -19.59745 1000.0 1.0 -19.59745 0.016 1.4 0.0 1.0 1.0 0.0 1.0

L M R

L M R

L M R

0 0.1 0.1 0.9 0.9 1.0

1.0 1.0 1.0

0.0 0.0 0.0

1000.0 0.01 100.0

x x x

u u u

p p p


40/58

Test 1 (sonic flow). Exact (line) and numerical solution (symbols) using

two numerical schemes (TV and TV-AWS) for the flux splitting of this

paper.


41/58

Test 1 (sonic flow). Exact (line) and numerical solution (symbols) using

two numerical schemes: Liou-Steffen (LS) and Zha-Bilgen (ZB-orig).


42/58

Test 2 (low density). Exact (line) and numerical solution (symbols)

using two numerical schemes (TV and TV-AWS) for the flux splitting of

this paper.


43/58

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

this paper.


44/58

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

the flux splitting of this paper.


45/58

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

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


46/58

Test 6 (isolated stationary contact discontinuity). Exact (line) andnumerical solution (symbols) using two numerical schemes: the Zha-

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

Godunov-type numerical approach (ZB-God).


47/58

Test 7 (Woodward and Colellar blast wave problem). Referencesolutions (WAF and Godunovs method with exact Riemann solver) and

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

and TV-AWS (bottom).


48/58

Other Potential Schemes forthe Pressure System


49/58

Lax-Friedrichs

FORCE

Rusanov

HLL

Godunov Centered


50/58

Applications and Extension of the Scheme


51/58

General Equation of State

For the Euler equations for general equation of state

= , (85)the flux may be decomposed as

=

=

0

,(86)

with the corresponding advection and pressure flux

=

, =

0

. (87)


52/58

The advection system is = 0,(88)This must be solved to :

1) Find advection speed to be used in the advection numerical

oprerator+/2) Find the pressure operator +/


53/58

2

1 1

,

... ...

... ...

ll

m m

u

u puuvv

uww

E u E pQ F Qq uq

q uq

q uq

(89)


54/58

2

1

0

0

0

1/ 2,

0

... ...

0

... ...

0

l

m

u p

v

w

u e puA Q P Q

q

q

q

(90)

With the corresponding advection and pressure flux defined as


55/58

1 1 11 1

2

1 1 1 11 1 1 1

1 1 1 11 1 1 1 2

2 2 2

2 2 22 2

212 2 2 2 2 2 2

1 22 2 22 2 2 2

0 0

0

0

0 0

00

0

0

t x x

u

u pu p

u E pE p u

u

u

pu u p

p uE u E p

(91)

: density, : velocity, : pressure, : total energy,

: volume fraction, with +=1 (93): specific energy, given by an EOS, e.g.= , . (94)

21/ 2k k k k E u e (92)

u


56/58

1 1 1

1 1 1 1

2

1 1 1 1

2 2 2

2 2 2 2

2

2 2 2 2

1

2

0

1

2

u

u u

u u

A

u

u u

u u

1 1

1 1 1 1 1 1 1 1 1 2 1

2 2

2 1 2 2 2

2 2 2 2 2 2 1 2 2 2 2

0

0

x

x x x

x

x x

x x x

p

u e u p p u u

P u

p p p

u e u p p u p

(95)


57/58

Explicit/Implicit Version

Could separate slow advetion from fast pressure waves

Treat slow advection explicitly

Treat fast pressure waves implicitly

Could capture interfaces with explicit sheme at CFL unity.

Potential improvements.


58/58

New flus 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

Potential applications: general equation of state

multicomponent flow

two-phase flow

Extension of the scheme: explicit for advection and implicit forpressure terms

On Flux Vector Splitting for the Euler Equation

Documents

Transcript of On Flux Vector Splitting for the Euler Equation