Graziani F. (editor) Computational Methods in Transport

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Non-LTE Radiation Transport in High Radiation Plasmas 325

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Finite-Diﬀerence Methods Implemented

in SATURN Complex

to Solve Multidimensional Time-Dependent

Transport Problems

R.M. Shagaliev, A.V. Alekseyev, A.V. Gichuk, A.A. Nuzhdin,

N.P. Pleteneva, and L.P. Fedotova

Numerical simulation of multidimensional particle transport problems falls

into the category of the most complex and labor-intensive application prob-

lems.

The paper brieﬂy describes the numerical methods implemented in

SATURN codes to solve time-dependent problems. The codes are intended for

solving 2D and 3D both linear and nonlinear spectral transport problems in

the physics of high densities and high energies in multigroup approximation:

The system of multigroup neutron–transport equations written in cylin-

drical coordinates:

∂

∂t





+ LN

+ α

= F

⎛

⎝



j=1

+ Q

⎞

⎠

4π

∂

r∂r



1 − µ

cos ϕ · N



∂

∂z

(µN

∂

∂Φ



1 − µ

sin ϕ · N



−

∂

∂ϕ



1 − µ

sin ϕ · N





−1

dµ

2π



dϕ; i =1, 2,...,i1

The system of multigroup radiation transport equations in cylindrical in

cylindrical coordinates:

∂ε

∂t

+ Lε

+ χ

·i

2π



j=1

(0)

Here the transport operation is

Lε

= µ

∂ε

∂z

∂

∂r



r ·

1 − µ

· cos ϕε



−

∂

∂ϕ



1 − µ

· sin ϕε



328 R.M. Shagaliev et al.

V = V(r,z,Φ,t) – velocity;

= α(r,z,Φ,t) – collision factor;

β = β(r,z,Φ,t) – particle multiplication constant;

Q=Q(r,z,Φ,t) – independent source of par t icles;

()

ϕµ

,Ω=Ω

– a unit vector in the neutron flight direction;

N=N(r,z,Φ,t,µ,ϕ) – flux of neutrons at point (r,z,Φ,t) flying in

a given direction;

= cos θ (–1 < µ < 1, 0 < ϕ < 2π).

Ω

Energy equation:

∂E

∂t



i=1

(0)

∆ω

−



i=1

∆ω

−



i=1



j=1

(0)

∆ω

= ε

(r, z, µ, ϕ, ω

,t) – radiation intensity function; χ

= χ

(ρ, T

,ω

) – absorption cross-section;

– mean energy of photons in group i; χ

= χ

(ρ, T

,ω

) – scattering cross-section;

∆ω

– the range width in energy variable; χ

·i

= χ

+ χ

– full cross-section;

T = T(r, Z) – temperature of medium; ε

= ε

,ω

). – Planck function;

ρ – densiy of medium; e

(0)



−1



dµdf

= E

(ρ, T

) – internal energy;

– coeﬃcients describing energy exchange Q

(r, z, ω

) – independent source.

during Compton scattering.

The main ideas of the numerical method implemented in SATURN to sole

multidimensional time-dependent transport problems are:

1. Approximation to the transport equation in space variables is constructed

using non-orthogonal space grids; in 2D cases:

– Using regular non-orthogonal grids of convex quadrangles (Fig. 1);

Fig. 1. A structured quadrangular (regular) grid

Finite-Diﬀerence Methods Implemented in SATURN Complex 329

Fig. 2. Arbitrary polygons are components of a grid

– Using irregular non-orthogonal grids of arbitrary-shape convex polygons

(Fig. 2).

2. The following various conservative ﬁnite-diﬀerence schemes depending on

the category of problems to be solved are used for the grids above to

approximate transport equations:

– scheme with extended template;

– scheme with introduction of closing relations basing on moment equa-

tions;

– scheme based on the use of adaptively reﬁned grids in phase space.

The schemes above have the following common feature: with the use of

non-orthogonal grids they preserve important features of DS

– schemes,

such as the transport equation approximation within a single phase space

cell and, consequently, a possibility to resolve systems of grid equations

using sweep (point-to-point) method of computations. Nevertheless, they

diﬀer from each other in the accuracy of approximation using essentially

non-orthogonal grids, in monotone behavior of grid solution and some

other features.

3. The method of source iterations is used to solve the system of multiple-

group grid transport equations numerically. The following methods are

used to accelerate the iterative process convergence:

– FCA method for the category of time-independent linear multiple-group

problems of calculating the critical parameter K

eﬀ

;

– KM method for the category of time-dependent nonlinear multiple-

group problems of radiation transport.

330 R.M. Shagaliev et al.

P˝

Fig. 3.

4. The eﬃcient parallelization algorithms have been developed and success-

fully used for numerically solving transport equations on multiprocessors:

– for 2D geometry – combined algorithm with ﬁne-grain parallelization in

space variables;

– for 3D geometry – pipeline-type algorithm.

Finite-diﬀerence schemes used to approximate 2D transport equations us-

ing non-orthogonal space grids

Extended-template scheme. The scheme is built using the grid function

values in a cell, on sides and at vertexes of a quadrangular cell (Figs. 3, 4).

(2)

(12)

(1)

(23)

(34)

(3)

(4)

Fig. 4.

Finite-Diﬀerence Methods Implemented in SATURN Complex 331

1 Multiple-Group Transport Equation Approximation

• in time: implicit scheme with weighting multipliers;

• in angular variables: method of discrete ordinates (s

-quadratures);

• in space: using the extended template for non-orthogonal space grids

The extended template scheme features:

• the scheme is conservative;

• convergence to the transport equation solution with the second order of

accuracy using non-orthogonal space grids;

• the scheme complies with the diﬀusion limit condition in optically dense

media;

• it uses DSn-method quadratures to approximate the transport in angular

variables.

The system of grid equations includes the following:

1. Grid equations for particle balance in the grid cells:

∆V

n+1

− N

∆ t



l=1

P (l,l+1)

· N

n+γ

P (l,l+1)

+ R

n+γ

− R

q−1

n+γ

q−1

+ α

· N

n+γ

· V

=∆V

· F

where

P (l, l+1)

P (l)

+ r

P (l+1)





P (l+1)

− r

P (l)



−

1 − µ



P (l+1)

− z

P (l)



cos ϕ



l =1, 2, 3, 4 ,

=∆S

1 − µ

· sin ϕ

q−1

∆ϕ

q−1

=∆S

1 − µ

sin ϕ

∆ϕ

2. Additional correlations in time variable and angular variable ϕ

n+γ

= γN

n+1

+(1− γ) N

, 0.5 ≤ γ ≤ 1

= ηN

+(1− η) N

q−1

, 0.5 ≤ η ≤ 1

3. Additional space relations between the values of the desired function in a

cell, on sides and at nodes of a cell. The number of relations in the scheme

considered depends on the number of illuminated (exposed to particles)

quadrangular cell’s sides. Here the following three variants shown below

in Fig. 5 are possible depending on the values of µ

,ϕ

q, m

The following additional correlations correspond to the three illumination

variants in the extended-template scheme:

332 R.M. Shagaliev et al.

Ω

(a) (b)

(c)

Fig. 5. Examples of a cell exposure to particles (illumination)

(a) N

P (2,3)

= δN

P 3

+(1− δ) N

= δN

P 3

+(1− δ) N

P (3,4)

=0.5(N

P 3

+ N

P 4

) ,N

= δN

P 4

+(1− δ) N

P (4,1)

= δN

P 4

+(1− δ) N

P 1

(b) N

P 6

=0.5(3δ − 1) N

P 3

+0.5(1− δ)(N

P 5

+ N

P 2

+ N

P 4

)

= δN

P (3,4)

+(1− δ) N

P (1,2)

Below given are the results for one 2D test problem demonstrating the

extended-template scheme accuracy during numerical solution of problems

using essentially non-orthogonal space grids.

One-group time-independent transport equation speciﬁed in a cylinder

with dimensions 0 = R =1, 0=Z = 2 is considered in (r, z) geometry. Full

cross-section α and multiplication coeﬃcient ß are α =1.34 and ß = 2.25

respectively. The boundary condition is a zero incoming ﬂow. The proper

parameter λ has to be calculated.

Numerical computations for the problem were carried out in 24 directions

of particle ﬂights (6 ranges in µ). The following three methods were used to

choose a space grid and its further reﬁnement in order to study the scheme

convergence rate:

Method 1. Rectangular uniform grids with steps

= h

=1/7, 1/14,

1/21, 1/28 were chosen.

Method 2. Some initial essentially non-orthogonal and non-uniform grid was

constructed (Fig. 6). The subsequent grids were obtained by re-

ﬁnement of the initial ones with preservation of a “bad” form of

cells (angles of quadrangles remained the same).

With reﬁnement using method 2, evenness of the numbers of rows and

columns was provided, they were taken equal to 8, 14, 20, and 28. The Table 1

below shows the results of computations. It is clearly seen that the calculated

values of λ satisfy the formula corresponding to the second-order convergence:

λ =0.1647 −

,whereA becomes equal to 1 or 1.16 using method 1, or

method 2 of grid selection and reﬁnement, respectively.

Finite-Diﬀerence Methods Implemented in SATURN Complex 333

Fig. 6. A non-orthogonal grid 7 × 7

Table 1.

Number of Rows

and Columns

(method 1)

Rectangular

Grids

(method 1)

Non-orthogonal

Grids

(method 2)

Number of Rows

and Columns

(method 2)

7 × 7 0.153134 0.154008 8 × 8

14 × 14 0.161915 0.160656 14 × 14

21 × 21 0.163499 0.162438 20 × 20

28 × 28 0.164045 0.163346 28 × 28

1.1 Scheme with Introduction of Closing Relations Basing

on Moment Equations

During such scheme construction the solution inside a grid cell is represented

in the form of bilinear decomposition in space variable and in angular vari-

able ϕ :

N(τ,η,ϕ



,µ) ≈ N





τη

τϕ



τϕ



× +

ηϕ



ηϕ



where ϕ



∆ϕ

q−1/2

η, τ are the coordinates of a point of space inside a cell during bilinear

transformation of a quadrangle into a unit square (Fig. 7).

334 R.M. Shagaliev et al.

(-1,1)

(1,1)

(1,-1)

(-1,-1)

Fig. 7.

The balance equations for particles in space grid cells and the correspond-

ing moment equations are used to ﬁnd the unknown coeﬃcients of the ex-

pansion in series.

To construct the diﬀerence analogs to moment equations, the (1) is re-

placed by some simpliﬁed expansions in series.

1. Moment diﬀerence equation in angular variable f. To construct the equa-

tions of interest, we use the following representation of solution inside

the diﬀerence grid range in variable f : N(τ,η,ϕ



,µ) ≈ N



τϕ



τϕ



ηϕ



ηϕ





∆ϕ

q−1/2

. This expansion in series is apparently

obtained by omitting the terms without variable ϕ



in the original bilin-

ear expansion. Using the given expansion in series we construct a rather

simple grid moment equation in variable f that has the following form:

−6

∆ϕ

1 − µ

sin ϕ

− 6

∆ϕ

1 − µ

sin ϕ

q−1

+12

1 − µ

+12

∆ϕ

1 − µ



1 − µ



+12

∆ϕ

1 − µ

+ αV



ηϕ







− 12

1 − µ

∆ϕ

sin ϕ



ηϕ





+ J







−

1 − µ

∆ϕ

sin ϕ

τη

Finite-Diﬀerence Methods Implemented in SATURN Complex 335



1 − µ

∆ϕ



sin ϕdϕ



1 − µ





cos ϕdϕ







+ J

τη





1 − µ

∆ϕ





sin ϕdϕ



1 − µ



(ϕ



)

cos ϕdϕ





× (J

ηϕ



+ J



)+2r

α(J

ηϕ



+ J



)=12

1 − µ

∆ϕ

× sin ϕ

q−1

τϕ







− 12

1 − µ

∆ϕ

sin ϕ



τϕ











−

1 − µ

∆ϕ

sin ϕ

τη



1 − µ

∆ϕ



sin ϕdϕ



1 − µ





cos ϕdϕ







+ J

τη





1 − µ

∆ϕ





sin ϕdϕ



1 − µ



(ϕ



)

cos ϕdϕ





× (J

τϕ



+ J



)

+2r

α(J

τϕ



+ J



)=12

1 − µ

∆ϕ

sin ϕ

q−1



−1



cos ϕdϕ





−1



sin ϕdϕ



sin ϕ

− sin ϕ

q−1

∆ϕ

q−

; P

cos ϕ

q−1

− cos ϕ

∆ϕ

q−

;

ϕ =

(1 + ϕ



)ϕ

(1 − ϕ



)ϕ

q−1

2. Moment diﬀerence equations in space variables. Similarly to the case

described above, the simpliﬁed expansion in series containing only the

dependences on space bilinear variables τ, η is used to construct such

equations: