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Use of the Space Adaptive Algorithm to Solve 2D Problems 245

The initial energy corresponds to temperature T =0.0001 KeV.

The initial radiation intensity is ε(t, z, r, µ, ϕ) |

t=t0

=0.

Radiation propagation over the system is calculated. In view of one-

dimensional behavior of solution, the radiation temperature proﬁle along

R-axis at ﬁnal time is considered to be the result of computations.

• To study the eﬃciency of the space adaptive algorithm for solving the

equation of radiation transport and interaction with medium, the results

of computations using the conventional method (4, 8, and 9) and the results

of two computations with adaptivity (6 and 7) were compared:

• Computation 4. The space grid consists of 22 columns and 160 rows, time

step is ∆t =0.00002. The adaptive algorithm is not used.

• Computation 6. The space grid consists of 22 columns and 40 rows. The

adaptive algorithm of adaptability level 2 is used (maximum partition into

4 adaptive cells).

• Computation 7. The space grid consists of 22 columns and 20 rows. The al-

gorithm of adaptability level 3 is used (maximum partition into 8 adaptive

cells).

• Computation 8. The space grid consists of 10 columns and 20 rows. No

adaptive algorithm is used.

• Computation 9. The space grid consists of 10 columns and 40 rows. No

adaptive algorithm is used.

Figure 9 shows the results of computation 4 and computations 6–9.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

101 101.05 101.1 101.15 101.2 101.25 101.3 101.35

R, sm

, keV

Computation 4 Computation 6 Computation 7 Computation 8 Computation 9

Fig. 9. The radiation temperature proﬁle along R-axis at time t =0.05

246 A.V. Alekseyev et al.

Analysis of the results above shows that results of computations 6 and

7 with the use of the adaptive technique are almost the same, with a very

insigniﬁcant deviation from the result obtained using the reﬁned grid (com-

putation 4).

Figure 10 shows the space grid and the radiation temperature near the

radiation wave front for computations 6 and 7.

Fig. 10. The space grid along R-axis

Further, the eﬃciency of using the adaptive algorithm is discussed. The

time of computation using the reﬁned grid with no adaptation (160 rows,

computation 4) exceeds the time of computation on the grid of adaptability

level 2 (40 rows of the reference grid, computation 6) by a factor of 3.97 and

is 4.3 times more prolong as compared to the time of computation using the

grid of adaptability level 3 (20 rows of the reference grid, computation 7),

with the given precision of computations preserved.

More detailed consideration of the computational process shows that this

saving of computation time can be attributed to the fact that concentration

of the space grid (owing to the use of the adaptive technique) takes place near

the radiation wave front only. Computations for the remained sub-regions of

Use of the Space Adaptive Algorithm to Solve 2D Problems 247

the system are carried out using a more coarse reference grid. It means that

in computation 6 with regard to adaptive cells the numerical solution process

is actually carried out using the space grid of 45 rows, and in computation 7

the space grid of 35 rows is actually used.

5.3 Results of Computations for a 2D Problem

A rectangular region of length 5 cm and height 1.2 cm is considered in 2D

axial symmetry geometry (Fig. 11). The computational domain consists of

two physical regions: the 1

physical region is a dense casing {0 ≤ Z ≤ 5,

1 ≤ R ≤ 1.2}, the 2nd region is transparent {0 ≤ Z ≤ 5, 0 ≤ R ≤ 1}.

0.0

1.0

5.0

1.2

Physical region 1

Physical region 2

Fig. 11. The system geometry

The material density in the system is ρ = 1. The material energy depen-

dence on temperature is described by equation E = C

T ,whereC

=0.81.

Computations are carried out with consideration of photon absorption,

with the photon absorption cross-section being speciﬁed by the formula:

. The optically dense (the 1

physical region) and the optically trans-

parent (the 2

physical region) regions are speciﬁed using various values of

factor A.

In the 1

region (optically dense) A =50.89.

In the 2

region (optically transparent) A =0.1374.

The scattering cross-section was taken zero during computations (χ

=0).

The initial temperature T at all points of the system was assumed to be

0.0001.

The following boundary conditions for radiation are speciﬁed at the left

end of the rectangular region (boundary G3): the boundary condition “mirror

reﬂection” for the boundary section belonging to the dense casing and the

incoming isotropic radiation ﬂux corresponding to temperature T =1for

the boundary section belonging to the transparent region. Zero incoming

radiation ﬂux is speciﬁed at the upper boundary and the right end.

248 A.V. Alekseyev et al.

The result is that solution to the equation of photon transport and inter-

action with medium, for the given problem, has essentially 2D behavior.

A grid of 10 rows (5 rows per each physical region) and 50 columns

(NR = 10, NZ = 50) is taken as the reference grid in space variables. The

adaptive partition level is taken so as to provide the maximum partition cor-

responding to the grids in space variables with partition NR = 40, NZ = 200

and NR = 80, NZ = 400. Computation results – the material temperature

values along cross sections Z =2cmandZ =3cmattimet =0.01 – are

shown in Figs. 12, 13.

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1 2

NR=10, NZ=50

NR=20, NZ=100

NR=40, NZ=200

NR=60, NZ=300

Adaptive, NR=10(4), NZ=50(4)

Adaptive, NR=10(8), NZ=50(8)

Fig. 12. The material temperature values along line Z = 2 cm,t =0.01

In Fig. 12, the solution obtained on the grid NR = 10(8), NZ = 50(8)

using the adaptive technique appears to be close enough to the result of

computations on the grid NR = 40, NZ = 200 using the standard technique,

the adaptive computation time is 2.8 times less.

In Fig. 13, the solution obtained on the adaptive grid NR = 10(8),

NZ = 50(8) is close to the result of computations using the ﬁnest space

grid (NR = 60, NZ = 300), while the adaptive computation time is 4.8 times

less. The results obtained for a lower adaptability level also demonstrate that

computation on the grid NR = 10(4), NZ = 50(4) gives the solution close

to solution obtained on the grid NR = 40, NZ = 200, while the saving of

running time is 8.9 times.

The same adaptive computation (NR=10(4), NZ=50(4)) provides a sig-

niﬁcantly more accurate solution than the computation on the grid NR = 20,

Use of the Space Adaptive Algorithm to Solve 2D Problems 249

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1 2

NR=10, NZ=50

NR=20, NZ=100

NR=40, NZ=200

NR=60, NZ=300

Adaptive, NR=10(4), NZ=50(4)

Adaptive, NR=10(8), NZ=50(8)

Fig. 13. The material temperature values along line Z = 3 cm, t =0.01

NZ = 100 (see, especially, Fig. 13), however, even in this case the adaptive

computation is 1.4 times faster.

As an illustration demonstrating the results of operation of the adap-

tive grid generation codes, Fig. 14 shows the distribution of the radiation

temperature ﬁeld over the system at three times. Black lines correspond to

the reference space grid, and white lines correspond to the adaptive cells.

The maximum adaptability level in this computation is 2 (four adaptive cells

comprise a reference one) in the both directions. It is clearly seen that the

reference space grid is adaptively partitioned at the wave front only and the

partitions are diﬀerent in various spatial directions.

The results presented allow us to conclude, by the example of the 2D test

problem, that the adaptive algorithm of reﬁning space grids demonstrates a

noticeable increase in the eﬃciency of computations.

5.4 Statement of a Demo Problem and Results of Computations

This section gives the description of the problem solved to demonstrate op-

eration of algorithms using an adaptive grid.

A cylindrical layer (0.0 cm ≤ Z ≤ 4.0 cm; 100.0 cm ≤ R ≤ 103.0cm)

with the speciﬁed medium features is considered. The problem is solved in

one-group “gray matter” approximation.

The following parameters are speciﬁed for the computational domain:

• Coordinate Z:0.0cm≤ Z ≤ 4.0cm,

• Coordinate R: 100.0 cm ≤ R ≤ 103.0cm;

250 A.V. Alekseyev et al.

t=0.004

t=0.006

t=0.008

Fig. 14. The adaptive reﬁnement of the space grid NR = 10(4), NZ = 50(4)

• Density of material: ρ =1.0 g/cm3;

• Absorption cross-section: χ

,whereA =50.890585, T is temperature

of material;

• No dissipation;

• The dependence of internal energy on temperature is described as E =

0.81 ∗ T .

The system geometry is shown in Fig. 15.

The incoming unilateral radiation energy ﬂux isotropically distributed

over the solid body and correspondent to radiation temperature T

= 2 KeV

is speciﬁed for the section of the left boundary (Z = 0; 101 ≤ R ≤ 102)

(solid line in Fig. 15).

The incoming unilateral radiation energy ﬂux isotropically distributed

over the solid body and correspondent to radiation temperature T

= 1.8 KeV

is speciﬁed for the section of the upper boundary (R = 103; 2 ≤ Z ≤

4.0). The boundary condition “free surface” (the incoming ﬂux equals 0) is

speciﬁed for the remained boundaries. Therefore, the problem is essentially

2D and allows the adaptive technique capabilities to be clearly demonstrated.

Use of the Space Adaptive Algorithm to Solve 2D Problems 251

100

101

102

103

024Z

Fig. 15. The system geometry

The initial energy corresponds to temperature T = 0.0001 KeV.

The initial radiation intensity is ε(t, z, r, µ, ϕ) |

t=t0

=0.

The reference space grid consisted of 20 columns and 15 rows. Construc-

tion of adaptive grids of no more than the third order (maximum partition

into 8 cells in both space directions) was allowed during the computation.

Figures 16–18 show some fragments of the computation results at three

times. The radiation temperature distribution and lines of the space grid

cells are represented, with the reference space grid’s lines being shown in

black color and the adaptive space grids lines – in white color.

One can clearly see from the ﬁgures above that a more ﬁne space grid

resultant from operation of the adaptive algorithms is generated on the radi-

ation wave front only, i.e. in places of maximum solution gradients. The level

of reﬁnement is determined by the gradient value. After the wave has passed

and the appropriate parts of the system have been heated, the computation is

carried out the reference grid again, because the solution gradient disappears.

6 Conclusion

The comparative computation results for all the test problems described in

the paper conﬁrm that the method of adaptively reﬁned grids is quite service-

able. The results of computations using the adaptive scheme with dynamically

reﬁning a grid in areas of solution changes appear to be close enough to those

obtained using the standard scheme on signiﬁcantly ﬁner space grids, or to

the exact analytical solution.

252 A.V. Alekseyev et al.

Fig. 16. The initial spatial distribution of the radiation temperature

Fig. 17. The spatial distribution of the radiation temperature at intermediate time

Use of the Space Adaptive Algorithm to Solve 2D Problems 253

Fig. 18. The spatial distribution of the radiation temperature at ﬁnal time

The use of the adaptive scheme allows signiﬁcant reduction of the time

required for computation while providing the speciﬁed accuracy of numerical

solution. Thus, in various test problems saving of running time in compu-

tations using the adaptive scheme was by a factor of 3 to 9 (depending on

the mode of computation), as compared to computations using the standard

scheme. The maximum saving of running time has been achieved for the 2D

problem described with the adaptive algorithms used to reﬁne cells of the

reference grid in two space directions simultaneously. This provides signif-

icant reduction of the total number of cells along with preservation of the

reﬁned diﬀerence grid in the system’s sub-regions having the primary eﬀect,

at a given time, on the grid solution accuracy.

The results obtained allow us to conclude that the method developed

for solving the 2D transport equation by adaptively choosing a space grid is

eﬃcient enough. The use of the adaptive algorithms makes it possible to solve

problems with essentially less running times, as compared to the standard

scheme, though a comparable precision of solution results is achieved.

254 A.V. Alekseyev et al.

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