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AIAA 2003–0249Implicit Kinetic Schemes for theEuler and the IdealMagnetohydrodynamics EquationsRamesh K. AgarwalWashington University, St. Louis, MO 63130

H.S. Raharjaya ReksoprodjoWichita State University, Wichita, KS 67260

41st Aerospace Science Meeting & Exhibit6–9 January 2003/Reno, NV

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41st Aerospace Sciences Meeting and Exhibit6-9 January 2003, Reno, Nevada

AIAA 2003-249

Copyright © 2003 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

Implicit Kinetic Schemes for the Euler and theIdeal Magnetohydrodynamics Equations

Ramesh K. Agarwal �

Washington University, St. Louis, MO 63130

H.S. Raharjaya Reksoprodjo y

Wichita State University, Wichita, KS 67260

The implicit kinetic schemes, namely the Kinetic Flux-Vector Split (KFVS) and theKinetic Wave/Particle Split (KWPS) schemes are derived for the Euler and the idealMHD equations. The schemes are applied to compute the ow in a 1-D shock tubeand the 2-D ow�eld due to a cylindrical explosion with and without the magnetic �eld.For the ideal 2-D MHD equations, the homogeneity of the ux vector is achieved byemploying the approach due to MacCormack, and a Poisson solver is used at every time-step to enforce the solenoidal condition on the magnetic �eld. To the authors' knowledge,this is the �rst time that the implicit kinetic schemes have been formulated for the Eulerand the ideal MHD equations.

Nomenclature

A Jacobian matrixa a \dummy" variable (� 1)BGK Bhatnagar-Gross-Krook~B magnetic �eld~c thermal or peculiar velocity (� ~v � ~u)et speci�c total energy per unit massF ux vectorf probability density distribution functionht speci�c total enthalpy per unit massI identity matrixJ (f; f) collision integralJ Jacobian of coordinate transformationKFVS Kinetic Flux-Vector SplitKWPS Kinetic Wave/Particle SplitMHD Magnetichydrodynamicsm molecular massn microscopic number densityPo total pressure

�� p+ 12B

2k

�p thermal (hydrodynamic) pressureQ �eld vector~u uid velocity~v molecular velocity� equivalent temperature�

�� �

2p ; Euler equations�

2Po; MHD equations

� ratio of speci�c heats� internal energy�o average internal energy�

��

1 �1 � 3

2

�RT

��William Palm Professor of Mechanical Engineering, Depart-

ment of Mechanical Engineering, Fellow AIAAyGraduate Research Assistant, National Institute for Avia-

tion Research, Student Member AIAACopyright c 2002 by the authors. Published by the American

Institute of Aeronautics and Astronautics, Inc. with permission.

� generalized coordinate� density collision invariant vector�

� �1;v; I + 1

2v � v�T�

Introduction

IN recent years, there has been a considerable inter-est in the kinetic schemes for solving the Euler,1, 2

Navier-Stokes,3 and MHD4, 5 equations. However, todate, all the papers on kinetic schemes reported inthe literature employ an explicit formulation due to itssimplicity. The major drawback of the explicit formu-lation is the restriction placed on the time-step allowedfor the stability of the scheme.Kinetic schemes are based on the fact that the set

of equations governing the motion of uid ows at thecontinuum level, i.e., Euler, Navier-Stokes, and Bur-nett equations, can be obtained by taking the momentsof the Boltzmann equation at the molecular level withrespect to the collision invariants. This is often re-ferred to as the \moment method strategy". For gasin the state of collisional equilibrium, the collision in-tegral vanishes, and the Boltzmann equation adopts aform similar to that of the linear wave equation. Itssolution is simply the Maxwellian velocity distributionfunction. When moments of this equation are takenwith the collision invariants, the Euler equations areobtained.One of the well-known and extensively used kinetic

schemes, the Kinetic Flux-Vector Splitting (KFVS)scheme proposed by Mandal & Deshpande,1 splits the ux term in the Boltzmann equation into positive andnegative parts based on the sign of the molecular veloc-ity. Taking the moments of this split- ux with respectto the collision invariant vector results in the KFVS

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algorithm. This scheme, however, requires the evalua-tion of the computationally expensive error functions.Recently, Agarwal & Acheson2 have proposed a

new ux splitting at Boltzmann level, which they callKinetic Wave/Particle Splitting (KWPS). They haveshown that this new splitting does not lead to the eval-uation of error functions, which results in increasedcomputational speed and also simpli�es the algebraconsiderably. By recognizing that the molecular veloc-ity of an individual gas particles can be expressed asthe sum of the average uid velocity of the gas and eachparticle's thermal (peculiar) velocity, the Boltzmann ux can be split into two components: the convectivepart and the acoustic part. These two parts of theBoltzmann ux are then upwind discretized and themoments of this discretized equation with the colli-sion invariants then result in the KWPS scheme forthe Euler equations.The schemes mentioned above have been investi-

gated using explicit formulations. This necessitateslimiting the time step for the sake of stability. Im-plicit ow solvers do not su�er from this drawback,allowing larger time-step, and thereby signi�cantly re-ducing the computation time for a wide range of owproblems. Therefore, it is desirable to derive implicitformulations for the kinetic schemes, both the KFVSand the KWPS schemes.Recently, for the �rst time, implicit formulations of

two kinetic schemes, namely the Kinetic Flux-VectorSplit (KFVS) and the Kinetic Wave/Particle Split(KWPS) schemes were systematically derived by Rek-soprodjo & Agarwal.6 They showed that for steadystate calculations, the implicit kinetic schemes weresigni�cantly more e�cient than the explicit schemeswhile retaining the accuracy and robustness of explicitschemes.For the ideal 1-D MHD equations, the KFVS scheme

was �rst formulated by Croisille et al7 to solve the7-wave model. However, their scheme was not di-rectly obtained from the Boltzmann equation due tothe lack of appropriate distribution function. Thismethodology was also employed by Reksoprodjo &Agarwal4 in developing the explicit KWPS and KFVSschemes for the ideal MHD equations and by Xu5 forhis BGK-scheme. Later, Tang & Xu8 derived the8-wave multi-dimensional BGK-scheme for the idealMHD equations. It is interesting to note here thatrecently Huba & Lyon9 have presented a method ofobtaining the uid portion of the 8-wave ideal MHDequations directly from the molecular level by mod-ifying the Boltzmann equation and the Maxwelliandistribution function by including an acceleration termdue to the magnetic �eld.Implicit upwind-split schemes for the ideal MHD

equations are impossible to derive due to the non-homogeneity of the ux vectors with respect to thestate vector. By introducing a \dummy" equation,

MacCormack10 has been successful in recovering thehomogeneity of the ux vectors. Using this new ex-panded system of equations, he derived an implicitscheme based on the Steger-Warming ux-splitting al-gorithm. It should be noted that Powell's magneticdivergence wave11 is inherently included in this for-mulation. This methodology has been applied in thispaper to derive the implicit formulation of the kineticschemes for the ideal MHD equations. A Poisson solveris used at every time-step to enforce solenoidal condi-tion on the magnetic �eld.In this paper, the derivations of the implicit kinetic

schemes for the Euler and the ideal MHD equationsare presented and validated by computing several testcases in 1-D and 2-D. The test cases include the ow ina 1-D shock tube, the 1-D shock structure, and the 2-Dcylindrical explosion with and without the magnetic�eld.

Implicit Kinetic schemes for the Eulerequations

For a gas in a state of collisional equilibrium thecollision integral vanishes and the solution of the Boltz-mann equation is the Maxwellian probability densitydistribution function. The Boltzmann equation can bewritten as

@ (nf)@t

+ vi@ (nf)@xi

= J (f; f) = 0 (1)

The Maxwellian distribution function is given by

f =1�0exp

�� �

�0

���

�

�3=2

exp���c2k� (2)

When moments of the Boltzmann equation (1) withthe Maxwellian distribution function (2) are takenwith respect to the collisional invariant vector

=�m mvj m�+ 1

2mv2k�T

the Euler equations are obtained. The moment of avariable ' with a weighting function f is de�ned asthe following mapping operation:

h'i =ZR+

d�ZR3

d3v ('f)

To obtain the implicit kinetic scheme for the Eu-ler equations, an implicit algorithm for the Boltzmannequation is formulated. The ux terms are linearizedas follows:

(vinf)t0+�t � (vinf)

t0 +�t@ (vinf)@Q

@Q

@t

� vi (nf)t0 + vi

@ (nf)@Q

�Q

The ux Jacobians are then obtained as

Ai =

�vi@ (nf)@Q

�2

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The KFVS methodology �rst splits the molecularvelocity v into positive and negative spaces, and thentakes the moments with respect to the collision invari-ant vector. Equation (1) is written as:

@f

@t+

@

@xi

�vi�jvij

2 f�= 0 (3)

On the other hand, the KWPS methodology �rst ex-presses the molecular velocity as the sum of the aver-age uid velocity and each particle's thermal velocity(v = u+ c), so that equation (1) becomes:

@f

@t+

@

@xi

�ui�juij

2 f + ci�jcij2 f

�= 0 (4)

After taking the moments of equations (3) and (4),the split- ux Jacobians are obtained. For brevity,they are written as the following matrix product A =BC�1, where

C =

24 1 0 0

uj �I 012u

2k �ul

1 �1

35

For KFVS scheme

A�� =k�kJ

�1�erf

�u�p

��

2 B1� � exp(�u2��)

2p

��B2�

�C�1 (5)

where

B1� =

264 u� ��l 0

u�uj �uj �l + �u�I �j12u�u

2k �ht�l + �u�ul

�1u�

375

B2� =

24 1 0 0

uj �I 012u

2k �ul

1 �1 +

12

35

For KWPS scheme

A�� =k�kJ

�u��ju�j

2 B0� +

12B

1� � 1

2p

��B2�

�C�1 (6)

where

B0� =

24 1 0 0

uj �I 012u

2k �ul

1 �1

35

B1� =

24 0 ��l 0

0 �uj �l �j0 �ht�l u�

35

B2� =

24 1 0 0

uj �I 012u

2k �ul

1 �1 +

12

35

Implicit Kinetic schemes for the idealMHD equations

By adding a \dummy" equation (@ta = 0) Mac-Cormack10 has obtained the homogeneity of the ideal

MHD ux-vectors, achieving the construction of an im-plicit upwind split- ux algorithms to be constructed.Since there is no distribution function available fromwhich the complete 8-wave MHD system can be ob-tained using the moment method strategy, the implicitkinetic schemes for the ideal MHD equations are ob-tained directly from the explicit formulations. For theKFVS scheme this process is very straightforward. Itssplit ux-vectors can be written as

F�� =k�kJ

�1�erf

�u�p

��

2 G1� � exp(�u2��)

2p

��G2

�

�(7)

where

G1� =

266666664

�u�

�u�uj + �jPo ��B�

a

�Bj

�u�et + u�Po ��B�

a

�ukBk

u�Bj ��B�

a

�uj a

0

377777775

G2� =

26666664

��uj

�et + 12Po �

�B�

a

�12B�

Bj ��B�

a

��j a

0

37777775

The split- ux Jacobians for the KFVS scheme canbe easily obtained from the above split- ux-vectors.Again, the matrices are written as the matrix productA = BC�1 where

C =

266664

1 0 0 0 0uj �I 0 0 012u

2k �ul

1 �1

1aBl � 1

2a2B2k

0 0 0 I 00 0 0 0 1

377775

The expressions obtained for the split- ux Jacobiansfor the KFVS scheme are

A�� =k�kJ

�1�erf

�u�p

��

2 B1� � exp(�u2��)

2p

��B2�

�C�1 (8)

where

B1� =

2666664

u� �l� 0u�uj �l�uj + �u�I �j12u�u

2k �l�ht + �u�ul � 1

aB�Bl

�1u�0 �lBj �B�I 00 0 0

0 01a �jBl � 1

aB�I � 12a2 �jB

2k

2au�Bl � 1

aB�ul � 1a2u�B

2k

u�I � 1aB�uj

0 0

377775

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B2� =

266664

1 0 0uj �I 012u

2k �ul

1 �1 +

12

0 0 00 0 0

0 00 0

32aBl � 1

2a �lB� � 34a2B

2k

I � 1a �jB�

0 0

377775

However, for the KWPS scheme it is not simple.The split- ux Jacobians are obtained by comparisonwith those of the KFVS scheme using the split- uxJacobians for the Euler equations as guidelines. Thesplit- ux vectors for the KWPS scheme are

F�� =k�kJ

�u��ju�j

2 G0� � 1

2G1� � 1

2p

��G2

�

�(9)

where

G0� =

2666664

��uj�et

Bj ��B�

a

��j a

0

3777775

G1� =

266666664

0

�jPo ��B�

a

�Bj

u�Po ��B�

a

�ukBk

��B�

a

��uj � �ju�

�a

0

377777775

G2� =

26666664

��uj

�et + 12Po �

�B�

a

�12B�

Bj ��B�

a

��j a

0

37777775

Now by comparing with equation (6), the followingexpression is obtained for the split- ux Jacobians forthe KWPS scheme:

A�� = k�kJ

�u��ju�j

2 B0� +

12B

1� � 1

2p

��B2�

�C�1 (10)

where

B0� =

266664

1 0 0 0 0uj �I 0 0 012u

2k �ul

1 �1

1aBl � 1

2a2B2k

0 0 0 I � 1a �jB�

0 0 0 0 0

377775

B1� =

266664

0 �l� 00 �l�uj �j0 �l�ht � 1

aB�Bl u�0 �lBj �B�I 00 0 0

0 01a �jBl � 1

aB�I � 12a2 �jB

2k

1au�Bl � 1

aB�ul � 12a2u�B

2k

0 � 1aB�

�uj � �ju�

�0 0

3777775

B2� =

266664

1 0 0uj �I 012u

2k �ul

1 �1 +

12

0 0 00 0 0

0 00 0

32aBl � 1

2a �lB� � 34a2B

2k

I � 1a �jB�

0 0

377775

Note that when computing the split- ux Jacobians,

the terms involving @@Q

�B�

a

�can be dropped since

they are homogeneous of degree zero.

0 500 1000 1500 2000

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Flo

wV

aria

bles

DensityVelocityPressureMach Number

Fig. 1 Implicit KFVS calculations for the Eulershock-tube test case

0 500 1000 1500 2000

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Flo

wV

aria

bles

DensityVelocityPressureMach Number

Fig. 2 Implicit KWPS calculations for the Eulershock-tube test case

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0 25 50 75 100

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Flo

wV

aria

bles

DensityVelocityPressureMach Number

Fig. 3 Implicit KFVS calculations for the Eulershock-structure computations

0 25 50 75 100

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

Flo

wV

aria

bles

DensityVelocityPressureMach Number

Fig. 4 Implicit KFVS calculations for the Eulershock-structure computations

Numerical results and discussions

The performance of implicit kinetic schemes is eval-uated by computing a number of benchmark test casesin 1 and 2 dimensions. In particular, the Euler calcu-lations are performed for 1-D shock-tube and steadystate shock structure, and the MHD computations areperformed for 1-D MHD shock-tube.12 In 2-D bothEuler and MHD computations are performed for ow�eld due to a cylindrical explosion.Results are compared with analytical results and so-

lutions from explicit kinetic schemes. It is shown thatthe implicit kinetic formulations retain the e�ciencyand robustness of their explicit counterparts withoutthe restrictive time step constraints. This results inan increase of computational speed for steady statecomputations.The Euler and MHD shock-tube test cases are com-

puted over a grid of 2000 points divided equally, withdensity and pressure ratios of 8 and 10 respectively.A time step of �t = 0:4 () CFL � 0:877) is used inthe explicit schemes, and a value of �t = 8:0 is used

0 250 500 750 1000

10-24

10-19

10-14

10-9

10-4

101

Iter

atio

nH

isto

ry

Explicit KFVSExplicit KWPSImplicit KFVSImplicit KWPS

Fig. 5 History of convergence in the Euler shock-structure computations

0 200 400 600 800

-1.5

-1

-0.5

0

0.5

1

Flo

wV

aria

bles

DensityThermal PressureUxUyBy

Fig. 6 KFVS calculations for the ideal MHDshock-tube test case

in the implicit schemes. Final solution is obtained att = 400. The Euler shock-structure computations areperformed on a grid of 100 points for a Mach 1:5 ow.A linear variation of the state variables spanning halfof the domain is used for the starting solution. A timestep of �t = 0:3 () CFL � 0:890) is used in the ex-plicit schemes, and a value of �t = 3:0 is used for theimplicit schemes. Final solution is obtained after 1000iterations. The initial conditions for the MHD shock-tube test case is the same as in the Euler case, withthe addition of a constant longitudinal magnetic �eldvalue of Bx = 3

4 , and a tranverse magnetic �eld hav-ing the value of By = +1 in the high density/pressureregion and By = �1 in the low density/pressure re-gion. The time step is set to �t = 0:16 for the explicitscheme, which roughly corresponds to CFL � 0:608.Final solution is obtained at t = 80. Also, for theEuler computations = 7

5 , and for the ideal MHDcomputations = 2.Figs. 1{2 show the computations for the Euler

shock-tube obtained with implicit KFVS and KWPSschemes. Figs. 3{4 show the for steady state compu-

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0 200 400 600 800

-1.5

-1

-0.5

0

0.5

1

Flo

wV

aria

bles

DensityThermal PressureUxUyBy

Fig. 7 KWPS calculations for the ideal MHDshock-tube test case

tations for the steady state shock-structure obtainedwith implicit KFVS and KWPS schemes using theEuler equations, and Fig. 5 shows the convergencehistory. Computations using the explicit schemes arenot presented here but are given in Reksoprodjo andAgarwal.4 Figures. 6{7 show the computations for theideal MHD shock-tube. These computations are in ex-cellent agreement with the solutions reported by Brioand Wu.12

Fig. 8 Density contours for case #1

In 2-D, the schemes are employed to compute the ow�eld due to a cylindrical explosion with and with-out the magnetic �eld. The details of this test casecan be found in Tang & Xu.8 The grid is a 100� 100Cartesian grid. The initial variables are

~B =

(~0 case #1

10p� | case #2

Time step is �t = 0:01 with �nal solution is computed

Fig. 9 Thermal pressure contours for case #1

Fig. 10 Density contours for case #2

at t = 3:0.Figures 8 and 9 show the density and thermal pres-

sure contours for case #1, and �gures 10 and 11 showthe density and thermal pressure contours for case #2.Figures 12 and 13 respectively show the density andthermal pressure variation along the horizontal andvertical center lines for case #1. As expected the den-sity and thermal pressure pro�les along both centerlines are identical. Figures 14 and 15 respectively showthe density and thermal pressure variation along thehorizontal and vertical center lines for case #2. Due tothe applications of the magnetic �eld in the vertical di-rection, the cylindrical blast wave does not propagatesymmetrically in all directions, and therefore the ther-mal pressure decreases more rapidly in the horizontal

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Fig. 11 Thermal pressure contours for case #2

Center line

Den

sity

­0.5 ­0.25 0 0.25 0.50

0.5

1

1.5

2

2.5

3 HorizontalVertical

Fig. 12 Density variation along center lines forcase #1

direction than in the vertical direction as shown in �g-ure 15. Furthermore, �gure 14 shows much shorterpeaks along the horizontal center line compared to thevertical direction, which indicates weaker shock fronts.Figures 16 and 17 respectively show the magnetic pres-sure contours and �eld lines for case #2. Figure 18shows the variation of the magnetic pressure along thehorizontal and vertical center lines. It shows a sig-ni�cant increase in the horizontal direction. This maysuggest that the dissipated energy from the shockwavein the horizontal direction, whuich is perpendicular tothe magnetic �eld lines, is transferred to the magnetic�eld instead of being converted into thermal energy.From these �gures it is clear that the presence of the

magnetic �eld can greatly a�ect the ow �eld. Insteadof a circular shock front obtained from the Euler com-putations, the shock front becomes elongated along thedirection of the magnetic �eld while the uid motion

Center line

The

rmal

Pre

ssur

e

­0.5 ­0.25 0 0.25 0.50

2

4

6

8

10

12

14

16

18

20 HorizontalVertical

Fig. 13 Thermal pressure variation along centerlines for case #1

Center line

Den

sity

­0.5 ­0.25 0 0.25 0.50

0.5

1

1.5

2

2.5

3 HorizontalVertical

Fig. 14 Density variation along center lines forcase #2

in the normal direction is suppressed. These resultsshow good agreement with those of Tang & Xu.8

Conclusion

Implicit kinetic schemes have been developed for theEuler and the ideal MHD equations. Numerical testsshow that the implicit schemes retain the accuracyand robustness of the explicit schemes while increas-ing their e�ciency by reducing the total computationtime.

References1Mandal, J. and Deshpande, S., \Kinetic Flux Vector Split-

ting for Euler Equations," Computers and Fluids, Vol. 23, 1994,pp. 447{478.

2Agarwal, R. and Acheson, K., \A Kinetic Theory BasedWave/Particle Flux Splitting Scheme for Euler Equations,"AIAA Paper 95-2178, 1995.

3Chou, S. and Bagano�, D., \Kinetic Flux-Vector Splittingfor the Navier-Stokes Equations," J. Comp. Physics, Vol. 130,1997, pp. 217{230.

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Center line

The

rmal

Pre

ssur

e

­0.5 ­0.25 0 0.25 0.50

2

4

6

8

10

12

14

16

18

20 HorizontalVertical

Fig. 15 Thermal pressure variation along centerlines for case #2

Fig. 16 Magnetic pressure contours for case #2

4Reksoprodjo, H. S. R. and Agarwal, R., \A Kinetic Schemefor Numerical Solution of Ideal Magnetohydrodynamics Equa-tions with A Bi-Temperature Model," AIAA Paper 2000-0448,2000.

5Xu, K., \Gas-kinetic Theory Based Flux Splitting Methodfor Ideal Magnetohydrodynamics," ICASE Report 98-53, 1998.

6Reksoprodjo, H. S. R. and Agarwal, R., \Implicit KineticSchemes for Euler Equations," AIAA Paper 2001-2629, 2001.

7Croisille, J.-P., Khan�r, R., and Chanteur, G., \NumericalSimulation of the MHD Equations by a Kinetic-type Method,"J. Sci. Comput., Vol. 10, 1995, pp. 481{492.

8Tang, H.-Z. and Xu, K., \A High-order Gas-kinetic Methodfor Multidimensional Ideal Magnetohydrodynamics," J. Comp.Physics, Vol. 165, 2000, pp. 69{88.

9Huba, J. and Lyon, J., \A New 3D MHD Algorithm: TheDistribution Function Method," J. Plasma Physics, Vol. 61,1999, pp. 391�.

10MacCormack, R., \An Upwind Conservation Form Methodfor the Ideal Magnetohydrodynamics Equations," AIAA Paper99-3609, 1999.

Fig. 17 Magnetic �eld lines for case #2

Center line

Mag

netic

Pre

ssur

e

­0.5 ­0.25 0 0.25 0.50

5

10

15

20

25

30

35

40 HorizontalVertical

Fig. 18 Magnetic pressure variation along centerlines for case #2

11Powell, K., \An Approximate Riemann Solver for Magne-tohydrodynamics," ICASE Report 94-24, 1994.

12Brio, M. and Wu, C., \An Upwind Di�erencing Schemefor the Equations of Ideal Magnetohydrodynamics," J. Comp.Physics, Vol. 75, 1988, pp. 400{422.

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