Numerical Simulation of ThreeDimensional QuasiNeutral Gas …elizar/publications/2015CMMP.pdf ·...

ISSN 0965�5425, Computational Mathematics and Mathematical Physics, 2015, Vol. 55, No. 8, pp. 1330–1345. © Pleiades Publishing, Ltd., 2015.Original Russian Text © T.G. Elizarova, M.V. Popov, 2015, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2015, Vol. 55, No. 8, pp. 1363–1379.

1330

1. INTRODUCTION

In this paper we present a numerical algorithm and simulation results obtained for unsteady three�dimensional flows of an ideal quasi�neutral plasma in the field of electromagnetic forces. The algorithmis an extension of previously constructed finite�difference schemes based on the quasi�gasdynamic(QGD) equations for viscous compressible gas flows [1–3]. This work is closely related to the researchinterests of Professor Favorskii, who, as a renowned expert in the computational fluid dynamics, activelysupported a then nascent scientific area whose development has led to the formation of the QGDapproach. As the head of a scientific school in computational MHD, Professor Favorskii would haveappreciated the results presented below.

The QGD equations express the conservation laws for gasdynamic variables, namely, the density,momentum components, and energy averaged over a short time interval. It is assumed that an averagedquantity is a smooth function of time, which can be expanded in a Taylor series about every time t, i.e.,

. (1)

The conservation law for the averaged quantity must contain terms reflecting the conservation of and of a correction term proportional to a small parameter that has the dimension of time. Thus,

the QGD equations represent the Navier–Stokes equations involving additional dissipative terms. Theseterms play a stabilizing role in the numerical solution of the equations, since they introduce additional dis�sipation.

The QGD equations with allowance for magnetic fields were first considered in [3, 4] for the descrip�tion of viscous gas and fluid flows. The effect of a magnetic field was taken into account in the form ofmagnetic forces and dissipative �correction terms in the gas dynamics part, while the field itself wasdescribed by Maxwell’s equations without �correction. For this system, an entropy balance equation was

' '( )1( ) ( ) ( )

t

t

f r tf r t f r t dt f r t …

t

+τ

∂ ,, = , ≈ , + τ +

τ ∂∫

( )f r t,( )f r t, τ

τ

τ

In blessed memory of Professor A.P. Favorskii

Numerical Simulation of Three�Dimensional Quasi�Neutral Gas Flows Based on Smoothed Magnetohydrodynamic Equations

T. G. Elizarovaa and M. V. Popova, b

a Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047 Russiab École Normale Supérieure de Lyon, CRAL (UMR CNRS 5574), Université de Lyon 1,

46 allee d’Italie 69007 Lyon, Francee�mail: [email protected], [email protected]

Received January 26, 2015

Abstract—A new finite�difference method for the numerical simulation of compressible MHD flowsis presented, which is applicable to a broad class of problems. The method relies on the magneticquasi�gasdynamic equations (referred to as quasi�MHD (QMHD) equations), which are, in fact, thesystem of Navier–Stokes equations and Faraday’s laws averaged over a short time interval. TheQMHD equations are discretized on a grid with the help of central differences. The averaging proce�dure makes it possible to stabilize the numerical solution and to avoid additional limiting procedures(flux limiters, etc.). The magnetic field is ensured to be free of divergence by applying Stokes’ theorem.Numerical results are presented for 3D test problems: a central blast in a magnetic field, the interac�tion of a shock wave with a cloud, and the three�dimensional Orszag–Tang vortex. Additionally, pre�liminary numerical results for a magnetic pinch in plasma are demonstrated.

DOI: 10.1134/S0965542515080084

Keywords: magnetic quasi�gas dynamics, QMHD, MHD flows, finite�difference algorithm, central�difference approximation.

COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS Vol. 55 No. 8 2015

NUMERICAL SIMULATION 1331

constructed, the exact solution of the Hartman problem was obtained, and the flow of an electrically con�ducting melt was computed in the no induction approximation.

The above described averaging procedure can also be applied to the magnetic field equations written inthe framework of the unified MHD system. Due to this approach, magnetic viscous flows would bedescribed with the help of quasi�gasdynamic equations for magnetohydrodynamics (referred to as quasi�MHD (QMHD) equations) in a self�consistent form. Such equations were first considered in [5, 6], wherethey were examined as applied to standard 1D and 2D tests, namely, the Riemann problem, propagationof magnetic waves, the dissipation and decay of a Alfven wave, a blast wave in a magnetized medium, theOrszag–Tang vortex, and the interaction of a shock wave with a cloud. In all the cases, the numerical solu�tion was demonstrated to converge well to the exact solution in the case of mesh refinement.

In [7] the QMHD equations were extended to the case of the nonideal gas equation of state in the pres�ence of external forces and a heat source. A heat balance equation was derived, and the entropy propertiesof the QMHD equations were examined.

In this paper, we present the system of QMHD equations in the 3D case written componentwise anddescribe numerical results for 3D test problems, such as a central blast in a magnetic field, the interactionof a shock wave with a cloud, and the three�dimensional Orszag–Tang vortex. Additionally, preliminarynumerical results are demonstrated for a plasma pinch confined by a longitudinal magnetic field. The sta�bility of a pinch is a major problem in plasma confinement within magnetic traps.

2. QMHD EQUATIONS

The QMHD equations are written in a Cartesian coordinate system with use of the following standardnotation for independent variables: is the density; , , and are the velocity components; , ,and are the magnetic flux density components; and is the total energy per unit volume.

For the squared magnitudes of the velocity and magnetic flux density, we use the brief notation

The factor was included in the definition of the magnetic field . In this notation, the total energyper unit volume is written as

where is the specific internal energy. To make the system of equations closed, we need an equation ofstate. For the case of an ideal gas, it has the form where is the gasdynamic pressure and is the ratio of specific heats. The equation of state expressed in terms of temperature has the form

, where is the universal gas constant and is the mean molecular weight of the gas. Fromthis, the temperature is expressed as

A combination of , , and gives the total specific enthalpy:

The small parameter with respect to which we perform averaging is denoted by ; it has the dimensionof time (see (1)). For convenience, the second term in the Taylor expansion in (1) is denoted as the incre�ment :

Let us write out the increments of all the quantities to be used in what follows. These expressions arederived from the MHD equations for an inviscid non�heat�conducting quasi�neutral plasma [8]:

ρ xu yu zu xB yB

zB E

2 2 2 2x y zu u u u= + + ,

2 2 2 2x y zB B B B= + + .

/1 4π B

2 2

2 2

u BEρ

= ρε + + ,

ε

( 1)p = γ − ρε, p γ

p RT= ρ η R η

T T p R= η ρ .

E p ρ

( )H E p= + ρ.

τ

Δ

f f f= + Δ .

grad div1 1 1⎛ ⎞Δ = −τ − ,⎜ ⎟ρ ρ ρ⎝ ⎠u u

( )grad div grad divp

p p p⎛ ⎞Δε = −τ ε + , Δ = −τ + γ ,⎜ ⎟ρ⎝ ⎠

u u u u

grad21 1

2y i z ix i

i i

B B B BB BBu u pi x y z

∂ ∂⎡ ⎤∂⎛ ⎞⎛ ⎞∂Δ = −τ + + − + + ,⎜ ⎟ ⎜ ⎟⎢ ⎥ρ∂ ρ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠⎣ ⎦u

1332


ELIZAROVA, POPOV

where i = x, y, z and u is the velocity vector. The scalar product of the velocity with the gradient of andthe divergence of the velocity are given by

The mass conservation law in the framework of the QMHD system is written as

(2)

where the fluxes are corrected components of the mass flux density:

The correction value is proportional to and is written in terns of spatial derivatives, which are, in fact,the derivatives of fluxes in the Euler equation for MHD:

The conservation laws for the momentum components have the form

(3)

where i = x, y, z. The components of the tensor (i, j = x, y, z) express the force associated with the cor�rected momentum flux and the gasdynamic and magnetic pressure in each direction:

Here and below, the first and second indices are the column and row numbers, respectively. The tensor

(i, j = x, y, z) includes the Navier–Stokes viscous stress tensor , which is proportional to the dynamic

viscosity , and the tensor associated with QMHD correction terms proportional to :

(4)

( ) ( ) ( )i x i x i y i y i z i z iB B u u B B u u B B u u Bx y z

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

∂ ∂ ∂Δ = τ − + − + − ,

∂ ∂ ∂

f

grad x y zf f f

f u u ux y z

∂ ∂ ∂= + + ,

∂ ∂ ∂u

div y zx u uu

x y z

∂ ∂∂= + + .

∂ ∂ ∂u

0y zx j jj

t x y z

∂ ∂∂∂ρ+ + + = ,

∂ ∂ ∂ ∂

( )i i ij u w i x y z= ρ − , = , , .

τ

}22 2 ( ) ( )

2x x x y x y x z x z x

Bw u p B u u B B u u B Bx y z

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎧τ ∂ ∂ ∂= ρ + + − + ρ − + ρ − ,⎨ρ ∂ ∂ ∂⎩

}22 2( ) ( )

2y x y x y y y z y z y

Bw u u B B u p B u u B Bx y z

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎧τ ∂ ∂ ∂= ρ − + ρ + + − + ρ − ,⎨ρ ∂ ∂ ∂⎩

22 2( ) ( )

2z x z x z y z y z z z

Bw u u B B u u B B u p Bx y z

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎧ ⎫τ ∂ ∂ ∂= ρ − + ρ − + ρ + + − .⎨ ⎬ρ ∂ ∂ ∂ ⎭⎩

yi yizi zii xi xiT Tu T

t x y z x y z

∂ ∂Π∂ ∂Π∂ρ ∂ ∂Π+ + + = + + ,

∂ ∂ ∂ ∂ ∂ ∂ ∂

ijT

2 2

2 2

2 2

12

12

12

x x x y x y x z x z x

ij y y y z y z yx y x y

x z x z y z y z z z z

j u p B B j u B B j u B B

T j u B B j u p B B j u B B

j u B B j u B B j u p B B

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

+ + − − −

= − + + − − .

− − + + −

ijΠ

nsijΠ

µqmhdijΠ τ

ns qmhdij ij ijΠ = Π + Π ,



where

Note that and the diagonal terms are

Tensor in (4) is written as

To the increment , we apply differentiation rules according to which .

The magnetic filed equations have the form

(5)

where the tensor containing the electric field components is given by

(6)

while the tensor expressing the QMHD correction term is determined by a combination of increments:

(7)

The total energy equation is

(8)

where

Here, i = x, y, z and k is the thermal conductivity, which is determined in terms of , , , and the Prandtlnumber Pr:

4 2 23 3 3

4 2 2 .3 3 3

4 2 23 3 3

y yz zx x x

ns y y yz zx xij

y yz z zx x

u uu uu u u

x y z x y x z

u u uu uu u

x y y x z z y

u uu u uu u

x z z y z x y

∂ ∂∂ ∂∂ ∂ ∂⎛ ⎞− − + +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂

⎜ ⎟∂ ∂ ∂∂ ∂⎜ ⎟∂ ∂

Π = + − − +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂⎜ ⎟

∂ ∂∂ ∂ ∂∂ ∂⎜ ⎟+ + − −⎜ ⎟

∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ns nsij jiΠ = Π

( )div223

ns iii

ui x y z

i

∂Π = μ − , = , , .

∂u

qmhdijΠ

22

22

22

( )( ) ( ) ( )

2

( )( ) ( ) ( ) .

2

( )( ) ( ) ( )

2

x x x y x y x z x z x

qmhdij x y x y y y y z y z y

x z x z y z y z z z z

Bu u p B u u B B u u B B

Bu u B B u u p B u u B B

Bu u B B u u B B u u p B

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

Δ−ρ Δ − Δ − + Δ −ρ Δ + Δ −ρ Δ + Δ

ΔΠ = −ρ Δ + Δ −ρ Δ − Δ − + Δ −ρ Δ + Δ

Δ−ρ Δ + Δ −ρ Δ + Δ −ρ Δ − Δ − + Δ

Δ ab a b b aΔ = Δ + Δ

m mnm mnm mnyi yizi zii xi xiT TT TB T T

i x y zt x y z x y z

∂ ∂∂ ∂∂ ∂ ∂+ + + = − − − , = , , ,

∂ ∂ ∂ ∂ ∂ ∂ ∂

mij j i i jT u B u B= − ,

mn mij ij i j j i j i i jT T B u u B B u u B⎛ ⎞

⎜ ⎟⎝ ⎠

= Δ = Δ + Δ − Δ − Δ .

( )

( ) ( ),

y yz zx xxx x xy y xz z

yx x yy y yz z zx x zy y zz z

F QF QF QE u u ut x y z x y z x

u u u u u uy z

∂ ∂∂ ∂∂ ∂∂ ∂+ + + + + + = Π + Π + Π

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂+ Π + Π + Π + Π + Π + Π∂ ∂

( )2

2i i i x x y y z z

BF j H B u B u B u B⎛ ⎞= + − + + ,⎜ ⎟ρ⎝ ⎠

( ) ( )

2 1

.

i i i

i x x y y z z i x x y y z z

TQ k u u p Bi

u B B B B B B B B u B u B u

⎛ ⎞⎜ ⎟⎝ ⎠

∂= − + ρ Δε + ρ + Δ

∂ ρ

+ Δ + Δ + Δ − Δ + Δ + Δ

µ γ R

Pr( 1)

Rk

μγ= .

γ −

1334


ELIZAROVA, POPOV

Equations (2), (3), (5), and (8) make up the QMHD system. The parameters of the system are the ratioof specific heats , the dynamic viscosity , the Prandtl number Pr, the mean molecular weight of the gas ,and the small regularizing parameter . For rarefied gases, is naturally defined as

(9)

where l is the characteristic free path of gas molecules and is the maximum of the fastmagnetosonic velocity. For example, the fast magnetosonic velocity in the x direction is given by

where c is the speed of sound:

The proportionality constant in (9) is chosen in the range 0.1–0.4. Definition (9) means that is thetime over which a perturbation propagating at the maximum possible velocity travels a distance equal tothe free path in the gas.

In the case of rarefied gases, is related to as

where Sc is the Schmidt number.If the gas cannot be treated as rarefied, then, according to (9), . If the physical processes weakly

depend on the viscosity and thermal conductivity and the gas is not rarefied, then the QMHD equationspass into the standard MHD equation for an inviscid non�heat�conducting quasi�neutral gas. Written inconservative variables, these equations have the form (see [8])

where

γ µ η

τ τ

f

lc

τ = α ,

max x y zf f f fc c c c⎛ ⎞

⎜ ⎟⎝ ⎠

= , ,

2 22 22 2 21 1 4

2 2x xf

BB Bc c c c⎛ ⎞ ⎛ ⎞= + + + − ,⎜ ⎟ ⎜ ⎟ρ ρ ρ⎝ ⎠ ⎝ ⎠

c p= γ ρ.

α τ

τ µ

Scpμ = τ ,

0τ →

0t x y z

∂ ∂ ∂ ∂+ + + = ,

∂ ∂ ∂ ∂

U F G H

22 2

2

2

0

( )2

x

x xx

x y x yy

x z x zz

x

x y y xy

x z z xz

x x

u

Bu p Bu

u u B Buu u B Bu

Bu B u BBu B u BB

E Bu E p B

ρ⎛ ⎞⎜ ⎟ρ⎛ ⎞ ⎜ ρ + + − ⎟⎜ ⎟ρ ⎜ ⎟⎜ ⎟ ⎜ ⎟ρ −ρ⎜ ⎟ ⎜ ⎟⎜ ⎟ ρ −ρ ⎜ ⎟⎜ ⎟= , = ,⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ −⎜ ⎟⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎛ ⎞⎝ ⎠ ⎜ ⎟+ + − ⋅⎜ ⎜ ⎟ ⎟⎝ ⎝ ⎠ ⎠

U F

u B

22 2

22 2

22

22

00

( )( )22

y z

x y x y x z x z

y z y zy y

z zy z y z

z x x zy x x y

z y y z

y z z y

z zy y

u uu u B B u u B B

u u B BBu p BBu p B

u u B Bu B u Bu B u Bu B u B

u B u B

BB u E p Bu E p B

ρ ρ⎛ ⎞ ⎛⎜ ⎟ ⎜ρ − ρ −⎜ ⎟ ⎜

ρ −⎜ ⎟ ⎜ρ + + −⎜ ⎟ ⎜

⎜ ⎟ ⎜ ρ + + −ρ −⎜ ⎟= , =⎜ ⎟ −−

⎜ ⎟−⎜ ⎟

⎜ ⎟−⎜ ⎟

⎛ ⎞⎛ ⎞⎜ ⎟ + + − ⋅+ + − ⋅ ⎜ ⎟⎜ ⎜ ⎟ ⎟ ⎝ ⎝ ⎠⎝ ⎝ ⎠ ⎠

G H

u Bu B

.

⎞⎟⎟⎟⎟⎟

⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

⎠



For flows with low viscosity, for example, for ideal plasma flows, is a parameter of the numericalscheme. In this case, it plays an exclusively regularizing role and tends to zero as the cell size decreases.

Note that the QMHD equations do not contain terms proportional to the second time derivative, i.e.,

terms of the form . Formally, these terms can be written out when the original equations are aver�aged over time, but, in this paper, they are assumed to be small and are omitted (see [9]).

3. NUMERICAL ALGORITHM

The system of QMHD equations is solved numerically by applying an explicit difference scheme withderivatives approximated by central differences. We introduce a uniform grid that divides the computa�tional domain into cells of size , where denotes the corresponding mesh size. The indepen�dent variables are placed at the cell centers and are denoted by integer indices i, j, and k, which corre�spond to the x, y, and z directions. The half�integer indices denote quantities defined on cell interfaces.As an example, Fig. 1 shows two adjacent grid cells with quantities defined at their centers and on theinterface between them.

When Eqs. (2), (3), (5), (8) are solved numerically, the time and space derivatives are replaced by dif�ference expressions of the form

(10)

where is the time step, denotes the unknown quantities at the new time level , and denotesthe known quantities at the time level t. Rules (10) mean that the variations in the quantities within a cellare determined by the fluxes through its faces.

The time step is determined by the Courant condition

where is the Courant number, which is determined experimentally and is no higher than 1.

Since the spatial derivatives in (2), (3), (5), and (8) involve quantities that are themselves determinedby spatial derivatives, they are computed using differences between the centers of the adjacent cells:

This means that the unknowns placed at the cell center with index are computed using informa�tion from the adjacent cells with indices , , and . Thus, the stencil of thescheme consists of 27 points.

τ

/2 2tτ∂ ∂∼

x y zΔ × Δ × Δ Δ

1 2 1 2ˆ

i j ki j k i j k i j kff f ff f

t t x x

, ,, , + / , , − / , ,− −∂ ∂

→ , → ,∂ Δ ∂ Δ

tΔ f̂ t t+ Δ f

| | | | | |min

x y zi j k xi j k fi j k yi j k fi j k zi j k fi j k

yx ztu c u c u c, ,

, , , , , , , , , , , ,

⎧ ⎫⎪ ⎪ΔΔ ΔΔ = σ , , ,⎨ ⎬+ + +⎪ ⎪⎩ ⎭

σ

1

1 2

i j k i j k

i j k

f ffx x

+ , , , ,

+ / , ,

−∂= .

∂ Δ

f̂ ( )i j k, ,

( 1 )i j k± , , ( 1 )i j k, ± , ( 1)i j k, , ±

k

ji

Δy

Δz fi, j, k fi +1, j, k

fi +1/2, j, k

Δx

Fig. 1. Two adjacent grid cells with specified quantities.

1336


ELIZAROVA, POPOV

The stability of the difference scheme is ensured by the regularizing parameters , , and k, which areproportional to the cell sizes:

where

4. DIVERGENCE�FREE MAGNETIC FIELD

In the simulation of magnetic flows, it is necessary to ensure that the computed magnetic field is freeof divergence. For this purpose, we propose using the constrained transport method from [10]. The samemethod was used in [11] to construct a numerical scheme for computing MHD turbulence in an interstel�lar gas. The method is based on Faraday’ law of induction

(11)

where is the vector of electric field strength. In the QMHD approach, Eqs. (5) are replaced by Eqs. (11),in which the electric field components are written taking into account the regularizing terms.

The electric field strength in matter moving at the velocity u is given by

Therefore, according to (6) and (7), the electric field components on corresponding ñell boundaries(see Fig. 2) are written in view of the QMHD correction terms as

(12)

According to the constrained transport method, components (12) are transferred to the edge centers(see Fig. 3) using the formulas

τ µ

( 1)f

Rh pSc kc Pr

μγτ = α , μ = τ , = ,

γ −

2 2 2h x y z= Δ + Δ + Δ .

rott∂ ∂ = − ,B E

E

[ ].= − ,E u B

1 2 1 2 1 2

1 2 1 2 1 2 .

m mn m mn m mnxi j k yz yz xi j k zy zy yi j k zx zx

m mn m mn m mnyi j k xz xz zi j k xy xy zi j k yx yx

E T T E T T E T T

E T T E T T E T T

, + / , , , + / , , + /

+ / , , + / , , , + / ,

= + , = − − , = + ,

= − − , = + , = − −

( )1 2 1 2 1 2 1 2 1 1 2 1 1 2

1 4 1 2 3 4 1 21 2 1 4 1 2 3 4

14

8 8

zi j k zi j k zi j k zi j k zi j k

z z z z

i j k i j ki j k i j k

E E E E E

E E E Ey xy y x x

+ / , + / , + / , , + / , + , , + / , + , + / ,

+ / , + / , + / , + / ,+ / , + / , + / , + / ,

= + + +

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂Δ Δ+ − + − ,⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎝ ⎠

k

ji

Ex, Ez

i , j + 1/2, k

Ex, Ey

Ey, Ezi, j, k + 1/2

Fig. 2. Electric field components used in the magnetic field equations in the QMHD system.



where the corresponding derivatives are calculated depending on the velocity sign on the face,

with the use of difference expressions of the form

The resulting electric field components at the centers of cell edges are used in (11) to compute the mag�netic field components at the centers of cell faces at the next time level (see Fig. 4). The differenceapproximation of (11) is given by

otherwise

1 2

1 4

1 21 2 1 4 1 1 4

1 4 1 1 4

0

0

12

xi j k

i j k

zxi j k

i j k i j k

i j k i j k

Ez uy

EE z uyy

E Ez zy y

+ / , ,

, + / ,

+ / , ,

+ / , + / ,+ , + / ,

, + / , + , + / ,

⎧∂⎪ , > ,∂⎪

⎪∂⎪∂ ⎪

= , < ,⎨ ∂∂ ⎪⎪

⎛ ⎞∂ ∂⎪⎜ ⎟+ ,⎪ ∂ ∂⎜ ⎟⎪⎩ ⎝ ⎠

1 2

1 4

2 z i j k z i j kz

i j k

E EE

y y, + / , , ,

, + / ,

−∂ ⎛ ⎞= .⎜ ⎟

∂ Δ⎝ ⎠

t t+ Δ

( ) ( )1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 21 2ˆ xi j k zi j k zi j k yi j k yi j kxi j kt tB E E E EBy z

+ / , , + / , + / , + / , − / , + / , , + / + / , , − /+ / , ,

Δ Δ= − − + − ,

Δ Δ

( ) ( )1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 21 2ˆ yi j k zi j k zi j k xi j k xi j kyi j kt tB E E E EBx z

, + / , + / , + / , − / , + / , , + / , + / , + / , − /, + / ,

Δ Δ= + − − − ,

Δ Δ

k

ji

Cell (i, j, k)Ez i+1/2, j, k

Ez i+1, j +1/2, kEz i, j +1/2, k

Ez i+1/2, j +1, k

Fig. 3. Transfer of the electric field components from the centers of cell faces to the edge centers.

ji

k Ex i, j+1/2, k +1/2

Ez i �1/2, j +1/2, k

Ex i, j+1/2,k−1/2

Ez i+1/2, j +1/2, k

By i, j +1/2, k

Fig. 4. Computation of the magnetic field components at the time level t + Δt at the face centers according to Faraday’slaw of induction.

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ELIZAROVA, POPOV

The magnetic field components at the cell centers are calculated by simple averaging (Fig. 5):

(13)

The magnetic field given by (13) is free of divergence. This can be seen by calculating the divergence atcell vertices using the formula

5. NUMERICAL EXAMPLES

Below, we present the numerical results obtained by applying the method described to some popularMHD tests extended to three dimensions. The properties of the method as applied to one� and two�dimensional tests were analyzed numerically in [5, 6].

5.1. Blast in a Magnetic Field

The problem is to compute the structure of a perturbation caused by an excess pressure in a boundeddomain as the perturbation travels through a medium with a magnetic field (blast wave problem) [12]. Thecomputational domain was a cube with side , in which we introduced a uniform grid consisting of

cells. Initially, the space was filled with an ideal gas with , density , and pressure, except for the central spherical domain of radius , where the pressure was specified as

. In the x direction, a homogeneous magnetic field was applied (see Fig. 6). The compu�tations were performed with and Courant number .

The numerical results obtained at the time are presented in Fig. 7. Specifically, the loga�rithm of density is shown in the form of a three�dimensional distribution and the two�dimensional cross

( ) ( )1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 21 2ˆ yi j k yi j k yi j k xi j k xi j kzi j kt tB E E E EBx y

, , + / + / , , + / − / , , + / , + / , + / , − / , + /, , + /

Δ Δ= − − + − .

Δ Δ

( ) ( ) ( )1 2 1 2 1 2 1 2 1 2 1 21 1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ2 2 2xi j k xi j k xi j k yi j k yi j k yi j k zi j k zi j k xi j kB B B B B B B B B, , + / , , − / , , , , , + / , , − / , , , , , + / , , − /

= + , = + , = + .

( )

( )

( )

div 11 1 11 2 1 2 1 2

1 1 1 1 1 1 1 1

1 1 1 1 1 1 1 1

1ˆ ˆ ˆ4

1 ˆ ˆ ˆ ˆ4

1 1ˆ ˆ ˆ ˆ ˆ4 4

xi j k xi j kxi j k xi j ki j k

xi j k xi j k xi j k xi j k

yi j k yi j k yi j k yi j k yi j k yi j k

B BB Bx

B B B Bx

B B B B By y

, , , + ,+ , , + , + ,+ / , + / , + /

+ , , + + , + , + , , + , + , +

, + , + , + , , , + , , , + , + + , + ,

= − − −Δ

+ + − −Δ

+ + − − + +Δ Δ

B

( )

( ) ( )

1 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1

ˆ ˆ ˆ

1 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ4 4

yi j k yi j k

zi j k zi j k zi j k zi j k zi j k zi j k zi j k zi j k

B B B

B B B B B B B Bz z

+ , , + + , , +

, , + + , , + , , + , , , + , + + , + , + , + , + , + ,

− −

+ + − − + + − − .Δ Δ

1L =

120 120 120× × 1 4γ = . 1ρ =

1p = 0 05r = .

1000p = 10xB =

0 5α = . 0 1σ = .

23 10t −

= ×

By i,j −1/2, k

Bz i, j, k +1/2

Bx i −1/2, j, k

k

ji

By i, j +1/2, j, k

Bx i+1/2, j , k

i, j, k

Bz i, j, k − 1/2

Fig. 5. Computation of the magnetic field components at the time level t + Δt at the cell centers.



section along the plane with level lines. Due to the magnetic pressure directed orthogonally to thefield, the flow has a structure extended in the field direction. The maximum density is reachedin the areas of maximum medium resistance. Minimum values occur in the center: .

0y =

log 0 35ρ .∼

log 0 95ρ − .∼

γ = 1.4 ρ = 1 p = 1

d = 0.1 ρ = 1 p = 1000

L = 1

Bx = 10

Fig. 6. Blast in a magnetic field: initial conditions.

z

0

0.35

y x

Logρ

0.022−0.95 0.215

0.126

0.0370

−0.0519

−0.141

−0.230

−0.319

−0.408

−0.496

−0.585

−0.674

−0.763

−0.852

−0.941

Fig. 7. Blast in a magnetic field. The logarithm of the density is shown as a three�dimensional distribution and the two�dimensional cross section in the plane y = 0 with level lines at the time t = 3 × 10–2. Additionally, the values in typicalregions are indicated.

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ELIZAROVA, POPOV

0.05

0.050 0.3 1 x

1.00y

r = 0.15

ux = 11.25

γ = 5/3

ρ = 10

γ = 3.87p = 1Bx = 0

ρ = 1

Bx = 0

By = 0.56

Bz = 0.56

By = 2.18

Bz = –2.18

p = 1

Fig. 8. Interaction of a shock wave with a cloud: initial conditions.

xy

0.65

0.586

1.30.7

0.54

1.277

0.235

0.215

0.395

0.475

0.555

0.636

0.716

0.796

0.876

0.956

1.037

1.117

1.197

lоgρ

z

Fig. 9. Interaction of a shock wave with a cloud. The logarithm of the density is shown as a three�dimensional distributionand the two�dimensional cross section in the plane y = 0 with level lines at the time t = 6.09 × 10–2. Additionally, the valuesin typical regions are indicated.



5.2. Interaction of a Shock Wave with a Cloud

The problem is to compute the decay of a dense cloud interacting with a shock wave [13]. Thecomputational domain was a cube with side , in which we introduced a uniform grid consistingof cells. The shock wave was initiated by a discontinuity between two states

separated by the plane :

A cloud of density in a hydrostatic equilibrium with the surrounding medium was specified as aball of radius centered at the point (see Fig. 8). The computations wereperformed with , , and .

The numerical results obtained at the time are presented in Fig. 9. Specifically, the log�arithm of density is shown in the form of a three�dimensional distribution and the two�dimensional crosssection along the plane with level lines. The resulting configuration reflects the hydrodynamic flowpast an obstacle. Due to the resistance of the cloud to the flow, a hemispherical contact discontinuity arisesahead of the cloud. Behind the cloud, we see an anisotropic flow with a structure related to the directionof the magnetic field.

5.3. Three�Dimensional Orszag–Tang Vortex

This problem was first proposed in [14] in a two�dimensional formulation in order to study the evolu�tion of supersonic turbulence and, later, became known as the Orszag–Tang vortex. Initially, smooth ini�tial conditions are set, which rapidly give rise to a complex flow with the formation and interaction ofshock waves. The problem is hard to solve for many numerical schemes, since the arising gradients, whichare especially strong in the central part of the computational domain, can lead to oscillations and negativedensity values.

1L =

120 120 120× ×

( )x y z x y zu u u B B B p= ρ, , , , , , ,U 0 05x = .

(3 86859 11 2536 0 0 0 2 1826182 2 1826182 167 345)

(1 0 0 0 0 0 56418958 0 56418958 1)

L

R

U

U

= . , . , , , , . , − . , . ,

= , , , , , . , . , .

10ρ =

0 15r = . ( ) (0 3 0 5 0 5)x y z, , = . , . , .

0 5α = . 0 1σ = . /5 3γ =

26 09 10t −

= . ×

0y =

xy

z

1/2|B|2

0.225

0.208

0.191

0.174

0.157

0.139

0.122

0.105

0.0879

0.0707

0.0536

0.0364

0.0192

0.0020

Fig. 10. Three�dimensional Orszag–Tang vortex: the magnetic field energy at the time t = 0.5.

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ELIZAROVA, POPOV

p1 = 1, p1 =5 × 10−4

p0 = 0.01

p0 = 5 × 10−6

Bz

2p1

β��=

z

x

y

0.1

Fig. 11. Initial configuration of the plasma pinch in a magnetic trap.

x

z

1.61 × 10−4

1.5 × 10−4

ρ, 10−4

1.42

1.32

1.23

1.13

1.03

0.928

0.829

0.730

0.631

0.531

0.432

0.333

0.234

0.135

7.3 × 10−5

y

Fig. 12. Plasma pinch in a magnetic trap: density distribution with three characteristic levels 7.3 × 10–5, 1.5 × 10–4, and1.61 × 10–4 at some time.



Below, this problem was considered in a three�dimensional formulation for the first time. The compu�tational domain was a cube with side , in which we introduced a uniform grid consisting of

cells. Periodic conditions were set on the boundary. The following initial conditions wereproposed for the 3D case:

where . The computations were performed with α = 0.5, σ = 0.1, and .

The numerical results obtained at the time t = 0.5 are presented in Fig. 10. Specifically, the energy ofthe magnetic field is shown by level lines in the range of 0.02 to 2.25. The method produces a correct flowstructure with all discontinuities on an infinitely long time interval. As the computations continue, theflow gradually breaks up into small structures and its kinetic energy dissipates due to viscosity.

5.4. Confinement of a Plasma Pinch in a Magnetic Trap

The confinement of a hot plasma pinch in a trap with the help of a longitudinal magnetic field was sim�ulated. Such a pinch is known as a �pinch. A task of great interest from an engineering point of view, thisproblem concerns the creation of tokamak�type devices for thermonuclear fusion and has been addressedfor several decades (see, e.g., [15, 16]). In experiments, the time of pinch confinement in a stable state isabout several milliseconds, which is so short because of the development of hydrodynamic instability.We considered this process in a dimensionless formulation.

1L =

120 120 120× ×

/ /

0 0 0

25 (36 ), 5 (12 ),

sin(2 ) sin(2 ) sin(2 )

sin(2 ) sin(4 ) sin(4 )x y z

x y z

p

u z u x u y

B B z B B x B B y

ρ = π = π

= − π , = π , = π ,

= − π , = π , = π ,

/0 1 4B = π /5 3γ =

θ

y

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1.0x

−5.30

−5.22

−5.14

−5.06

−4.98

−4.90

−4.82

−4.74

−4.66

−4.58

−4.50

−4.42

−4.34

logρ

−4.26–3.8

Fig. 13. Development of hydrodynamic instability in the pinch computed in the 2D formulation in the xy plane on a400 × 400 grid. The density distribution at some time is shown on a logarithmic scale.

1344


ELIZAROVA, POPOV

The computational domain was a cube with side filled up with an ideal gas with (which

corresponds to hydrogen), density , and pressure . A pinch was set in the center of

the computational domain as a cylinder of radius with inside characteristics and

. Initially, we specified the longitudinal magnetic field , where . The initial config�uration of the pinch is shown in Fig. 11. A uniform grid consisting of cells was introduced.The computations were performed with and .

In the simulation, we obtained a cyclic process of plasma expansions and contractions under the influ�ence of the magnetic pressure. Uniformly distributed at the initial time, the magnetic field was pushed outof the pinch region. Several cycles of direct and reverse energy flow between the magnetic field energy andthe kinetic energy of the pinch matter were observed. The expansion of the plasma was suppressed byincreasing the energy of the field.

To verify the stability of the code performance, we computed a pinch inclined initially to the axis atan angle of . As a result, the pinch aligned out along the magnetic field via cyclic oscillations. Thisprocess is demonstrated in Fig. 12, which shows the density distributions at some time with three charac�

teristic levels: , , and . It can be seen that the shape of the pinch is deformedin the oscillations.

In the three�dimensional case, we failed to resolve small�scale structures and simulate the developmentof instability because of the rather coarse grid used. Such instability was obtained when the pinch edge inthe xy plane was simulated on a grid in the two�dimensional formulation. The instability devel�oped transversely to the field direction and resembled the Rayleigh–Taylor instability. The structures gen�erated in this process are demonstrated in Fig. 13, which displays the density distribution on a logarithmicscale at some time.

In the case of a weak magnetic field, no cyclic process or instability development was observed. Thepinch broke up due to diffusion.

6. CONCLUSIONS

An algorithm for solving the QMHD equations describing unsteady compressible MHD flows with theideal gas equation of state was presented. In contrast to earlier works, the QMHD equations were writtenis a self�consistent form within the framework of a unified approach to both conservation law equations influid dynamics and Faraday’s laws for magnetic fields.

The scheme represented is fully three�dimensional, and all physical quantities are computed withoutsplitting in space. The numerical algorithm was tested by computing several hard�to�solve three�dimen�sional MHD problems, on which the code exhibited an exceptionally stable performance. The chosennumerical parameter α = 0.5 and the Courant number σ = 0.1 are universal for any problem.

The algorithm is easy to implement but rather cumbersome because of the additional monotonizingcorrection terms proportional to . Due to the explicit scheme with central differences, the code was par�allelized in a natural way by applying domain decomposition over the processors.

An advantage of the approach is that there is no need to use conventional monotonizing procedures,such as limiters, which are required in the case of the standard MHD equations. Such monotonizing pro�cedures are not universal, so that the code has to be tuned in each particular case.

A shortcoming of the method is that it is first�order accurate, so finer grids have to be used to obtainsolutions comparable in quality to those produced by high�order schemes. Note that the order of accuracyis determined by the solution behavior in smooth domains, while the most interesting are discontinuoussolutions.

ACKNOWLEDGMENTS

This work was supported by the Russian Foundation for Basic Research, project no. 13�01�00773_a.This project has received funding from the European Union's Seventh Framework Program for

research, technological development and demonstration under grant agreement no. 320478.

REFERENCES

1. T. G. Elizarova, Quasi�Gas Dynamic Equations (Nauchnyi Mir, Moscow, 2007; Springer�Verlag, Berlin, 2009).

1L = /5 3γ =

60 5 10−

ρ = × 0 0 01p = .

0 1r = .

41 5 10−

ρ = ×

1 1p = /12zB p= β 5β =120 120 120× ×

0 5α = . 0 1σ = .

z10°∼

57 3 10−

. ×

41 5 10−

. ×

41 61 10−

. ×

400 400×

τ



2. B. N. Chetverushkin, Kinetic Schemes and Quasi�Gasdynamic System of Equations (MAKS, Moscow, 2004;CIMNE, Barcelona, 2008).

3. Yu. V. Sheretov, Continuum Dynamics under Spatiotemporal Averaging (NITs Regulyarnaya i KhaoticheskayaDinamika, Moscow, 2009).

4. T. G. Elizarova, I. S. Kalachinskaya, Yu. V. Sheretov, and I. A. Shirokov, “Numerical simulation of electricallyconducting liquid flows in an external magnetic field,” J. Commun. Technol. Electron. 50 (2), 227–233 (2005).

5. T. G. Elizarova and S. D. Ustyugov, Preprint No. 1, IPM RAN (Keldysh Inst. of Applied Mathematics, RussianAcademy of Sciences, Moscow, 2011).

6. T. G. Elizarova and S. D. Ustyugov, Preprint No. 30, IPM RAN (Keldysh Inst. of Applied Mathematics, RussianAcademy of Sciences, Moscow, 2011).

7. B. Ducomet and A. Zlotnik, “On a regularization of the magnetic gas dynamics system of equations,” KineticRelated Models 6 (3), 533–543 (2013).

8. A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov, Mathematical Aspects of Numerical Solution of Hyper�bolic Systems (Fizmatlit, Moscow, 2001; Chapman and Hall/CRC, London, 2001).

9. T. G. Elizarova, “Time averaging as an approximate technique for constructing quasi�gasdynamic and quasi�hydrodynamic equations,” Comput. Math. Math. Phys. 51 (11), 1973–1982 (2011).

10. T. A. Gardiner and J. M. Stone, “An unsplit Godunov method for ideal MHD via constrained transport in threedimensions,” J. Comput. Phys. 227, 4123–4141 (2008).

11. S. D. Ustyugov, M. V. Popov, A. F. Kritsuk, and M. L. Norman, “Piecewise parabolic method on a local stencilfor magnetized supersonic turbulence simulation,” J. Comput. Phys. 228, 7614–7633 (2009).

12. W. Dai and P. Woodward, “A simple finite difference scheme for multidimensional magnetohydrodynamicalequations,” J. Comput. Phys. 142, 331–369 (1998).

13. G. Tóth, “The ∇ ⋅ B = 0 constraint in shock�capturing magnetohydrodynamics codes,” J. Comput. Phys. 161,605–652 (2000).

14. S. A. Orszag and C.�M. Tang, “Small�scale structure of two�dimensional magnetohydrodynamic turbulence,”J. Fluid Mech. 90, 129–143 (1979).

15. A. A. Vedenov, E. P. Velikhov, and R. Z. Sagdeev, “Stability of plasma” Sov. Phys. Usp. 4, 332–369 (1961).16. V. E. Golant, A. P. Zhilinskii, and S. A. Sakharov, Fundamentals of Plasma Physics (Atomizdat, Moscow, 1977;

Wiley, New York, 1980).

Translated by I. Ruzanova

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