NEA C5G7-MOX benchmark obtainedwith the discrete ordinates code dort

Progress in Nuclear Energy, Vol. 45, No. 2-4, pp. 153-168, 2004 Available online at www.sciencedirect.com s c , ~ . c s ("-(~ o . . Bc 1-o ELSEVIER www.elsevier.com/locate/pnucene

© 2004 Published by Elsevier Ltd Printed in Great Britain 0149-I 970/$ - see front matter

doi: 10.1016/].pnucene.2004.09.007

RESULTS ON THE OECD/NEA C5G7-MOX BENCHMARK OBTAINED WITH THE DISCRETE ORDINATES CODE DORT

ANDREAS PAUTZ

TUV Hannover/Sachsen-Anhalt e.V. Am TIJV 1, 30519 Hannover, GERMANY e-mail: [email protected] 1

SIEGFRIED LANGENBUCH, ARMIN SEUBERT, WINFRIED ZWERMANN

Gesellschaft fiir Anlagen- und Reaktorsicherheit (GRS) mbH Forschungsinstitute, 85748 Garching, GERMANY e-mail: [email protected], [email protected], [email protected]

ABSTRACT The well-known multigroup Discrete Ordinates code DORT has been employed to perform calculations on the OECD/NEA C5G7-MOX benchmark. The participants were required to supply the effective multiplication factor as well as the pin power distribution of the problem specification. We show our submitted results to be well consistent with the reference solution, which was produced by Monte-Carlo calculations. Furthermore, we point out some improvements being made on the computational procedure, which give rise to an additional gain in accuracy. © 2004 Published by Elsevier Ltd

Keywords: C5G7; DORT; Discrete Ordinates; Benchmark; MOX

1. INTRODUCTION The C5G7-MOX benchmark is intended to be a measure of contemporary transport code capabilities to treat LWR core problems without spatial homogenization. It has been proposed in early 2001 by the Expert Group on 3-D Radiation Transport Benchmarks and is fully described in a recently published OECD/NEA report [1]. This report also contains a compilation of the submitted results as well as comparisons to the

1 New e-mail address: [email protected]

i53

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reference Monte-Carlo solution. Currently, a 3-D extended version of the benchmark with partially inserted control rods is in progress. We participated in this benchmark with the well-known SN-code DORT and its three-dimensional counterpart TORT [2] for the 3-D problem specification. Both codes solve the linear, first-order Boltzmann equation using the method of Discrete Ordinates, allowing for an (in principle) arbitrary number of energy groups and any desired scattering expansion (PN-) order. DORT and TORT have been used in a large variety of applications during the past two decades, e.g. for shielding analyses, criticality evaluations and also, as it has been demonstrated recently, in time-dependent problems [3]. A variety of spatial discretization methods is available in DORT, like the theta-weighted approach or the standard diamonddifferencing method (with and without zero-fixup). Moreover, DORT uses rather mature acceleration schemes for the inner iterations, particularly Coarse Mesh Rebalance (using either partial or global rebalancing currents) and Diffusion Synthetic Acceleration (DSA), which makes DORT very competitive for many classes of different problems. The outer iteration cycle can also be substantially accelerated by the so called error-mode extrapolation model, which uses the actual error decay rate to generate appropriate acceleration parameters. In addition, an upscatter rebalancing scheme is available to treat thermal groups more efficiently. To achieve the high accuracy desired in the C5GT-MOX benchmark, we have recompiled the DORT code under the Compaq Tru64 Unix, IBM AIX 5 and MS Windows operating systems, using double precision (i.e. 64-bit floating point) arithmetic, in contrast to the single precision version of DORT, as it traditionally comes in the DOORS package. This enables us to achieve a high degree of convergence in eigenvalues, fluxes and fission source distributions with virtually no penalty in CPU time (this statement holds at least for the 64-bit operating systems IBM AIX and Tru64 Unix).

2. SOLUTION METHOD

The solution of the C5G7-MOX problem requires a careful treatment of both the spatial and the angular variables. Since the participants were explicitly required not to use any spatial homogenization techniques, it was of paramount importance to construct a reasonable geometric model of the circular fuel rod embedded in the moderating region of the pin cell. Since DORT can only treat regular 2-D geometries (i.e. xy, r0 and rz), we had to approximate the fuel pin by a Cartesian mesh, the usual way being to define some "stair case" or "step" function, which describes the circular fuel perimeter sufficiently well. To illustrate this, we show in Fig. 1 (left) the original pin cell geometry, as it has been specified in the benchmark proposal and in Fig. 1 (middle) our approximate model. ~,y ] ~4

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155

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The main question using such a "stair-case" model is how to make an appropriate choice for the "step" coordinates, i.e. the abscissa/ordinate values ~0 to ~5 in Fig. 1 (right). There may be a large variety of reasonable methods to define these values. We were guided by the idea of strictly conserving the volumetric fractions inside the pin cell and to find some kind of geometric best-approximation to the unit circle, as shown in Fig. 1 (right). In the essence, this means that we minimized the gray shaded area between the actual quarter arc of the unit circle (dashed line in Fig. 1, right) and the stair case function (solid line in the same Figure). To achieve this, we constructed a special functional, which yields the "average distance" between points on the unit arc and the corresponding points (i.e. belonging to the same abscissa value) of the step function. The averaging of this quantity is done by integration. Let us define the functions:

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Table I. The step coordinates resulting from the variational procedure outlined in Eqs. 1 to 3 for one up to eight steps. 1 0.000000 0.886227

2 0.000000 0.536748 1.000000

3 0.000000 0.400733 0.780889 1.000000

4 0.000000 0.321452 0.628202 0.875520 1.000000

5 0.000000 0.268550 0.526487 0.753324 0.918891 1.000000

6 0.000000 0.230491 0.453302 0.656709 0.825076 0.942572 1.000000

7 0.000000 0.201761 0.397889 0.580791 0.741269 0.869406 0.957052 1.000000

8 0.000000 0.179313 0.354413 0.520017 0.670152 0.798369 0.898665 0.966606 1.000000

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thus giving a system of N+I non-linear equations. The solution of the resulting equation system (which we do not give here for brevity reasons) can easily be accomplished by standard numerical methods. For this purpose, we have used the IMSL-library routine NEQNF, which solves general non-linear equation systems by the so called modified Powell hybrid algorithm with a finite difference approximation of the Jacobian of the system. The iterative procedure yields values of the variational parameters xi, such that the side condition of volume conservation is fulfilled to better than 10 -7. For a step number N ranging from 1 to 8, we give the resulting coordinates in Table I above. The 1-step approximation of the fuel rod obviously corresponds to a square rod; therefore the boundary conditions of having 0.0 and 1.0 at either side cannot be fulfilled for this simplest case. After having constructed the step coordinates, the meshing for the C5G7-MOX problem can easily be generated. Since a pin cell with a N-step nodalization needs (2N+2)x(2N+2) meshes to be represented (cf. Fig. 1 (middle) for the 3-step nodalized pin cell), the dimensions of the problem grow rapidly with increasingly fine nodalization. For example, our 8-step nodalization corresponds to an 18x18 meshing per pin cell and thus requires 612 overall meshes in either spatial direction (the C5G7-MOX benchmark contains 34x34 pin cells) plus the meshes required in the surrounding moderator region. To limit the numerical effort, it is therefore interesting to investigate how many meshes are actually needed to appropriately approximate the circular fuel rod. Another important issue to be addressed in our calculations is the angular quadrature order to be applied to the problem as well as the nature of the quadrature set. For our submitted benchmark results, we have used quadrature sets for $2, $4, Ss and $16, which come with the DORT sample files in the DOORS package, as it is distributed by the NEA data bank or RSICC. However, as we found later, these quadratures (which originate from the early Discrete Ordinates code TWOTRAN) do not exactly obey the sum rules, which should usually apply to any Discrete Ordinates set. In a second run, we have therefore taken quadrature sets from the code TWODANT [4], which can generate very accurately the standard level-symmetric quadrature sets from $2 to $16 as well as the so called Chebychev-Legendre quadrature sets (which range from $4 to $50 in steps of two and are additionally implemented for $60, $70, $8o, $90 and $100).

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x

Chebychev-Legcndre

Quadrature

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The difference between the two different quadrature sets is shown in Fig. 2. Here we have depicted the angular half sphere (four octants of the unit sphere) of Discrete Ordinates directions, which are necessary to represent problems in two spatial dimensions. The two independent angular variables are usually called Ix and q. Most SN-codes group these angles together in q-levels, i.e. pairs with constant q and variable ~x (this simplifies the treatment of the so called angular redistribution terms present in Discrete Ordinates theory for curvilinear coordinates). In Fig. 2 we have depicted the angular set for an S6-quadrature, containing six ~q-levels. One can observe, that for the level-symmetric case (left picture), the q-levels closest to the poles of the unit sphere contain two angles only, while the q-levels near the equator have 6 angles. The overall number of directions necessary for an $6 calculation is therefore 24; in general N(N+2)/2 angular variables are needed for an SN-calculation with level-symmetric quadrature. In contrast, the Chebychev-Legendre quadrature (Fig. 2, right) has an equal number of angles for all qlevels, which means that particularly the Discrete Ordinates in the vicinity of the poles of the unit sphere are situated very close together, which has to be taken into account by low weighting factors for these directions. It is obvious, that the Chebychev-Legendre quadrature for $6 will require a larger numerical effort than the level symmetric one, since 36 angles are needed instead of 24; in general this quadrature set will require a number of N 2 angles for an SN-calculation. Finally, for both quadrature sets, DORT needs so called "starting directions" to initialize the angular sweep, which gives rise to an additional N directions

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with zero weighting factor, such that the overall number of angles is N2/2+2N respectively NZ+N for levelsymmetric and Chebychev-Legendre S~-quadrature.

3. B E N C H M A R K ~ S U L T S In what follows, we will briefly comment on our results and the parameter studies we performed in order to arrive at a reliable solution to the benchmark problem. In the benchmark specification, all participants were required to supply both eigenvalue and pin power distribution of the C5G7-MOX arrangement. Particularly the pin powers are a good measure of the quality of the solution and were checked very carefully. To give the reader an impression, how the pin power distribution looks like, we have sketched it in Fig. 3 as a bar plot over the xy-plane. The pin power has been normalized to an average of one fission per second per cell, resulting in a peak power of roughly 2.5 fissions per second per cell, at a position situated close to the assembly center. The pin power slightly increases, as one moves close to the reflector region, which is due to the enhanced moderation in this area. There is also a clearly visible effect in the vicinity of the interfaces between UO2 and MOX fuel, where a significant energy spectrum change takes place, the interfacial areas appearing quite pronounced in the plot of Fig. 3. Table II. Some relevant quantities, which have been compared to the reference solution. Quantit), Eigenvalue Maximum Pin Power Minimum Pin Power Assembly Power (Inner UOz) Assembly Power (Outer UO2) Assembly Power (MOX)

DORT Results 1,184818 2,510 0,232 494,5 i39,5 211,0

Reference Results 1,186550 2,498 0,232 492,8 139,8 211,7

Deviation -0,146% 0,48% 0,02% 0,34% -0,20% -0,34%

Table III. The error measures for our DORT solution, as they have been derived from the comparison to the reference solution. Average Error (AVG) 0,35%

RMS error 0,41%

MRE error 0,34%

Maximum Error 1,21%

In Tables II and III we list several quantities, which have been selected for comparison and which were extracted from the submitted results. Table II gives several important data on maximum and minimum pin power values, while Table III lists the error measures, which have been defined in the course of this benchmark. Our submitted results were produced using a very fine spatial discretization (8-step approximation, i.e. 18x18 meshes per pin cell) and a level-symmetric quadrature set of $16 which comes with DORT. The results we obtained came in many respects very close to the reference solution. To learn more about the influence of spatial and angular approximation, we performed numerous calculations with varying spatial meshes and the improved quadrature sets taken from TWODANT. Figure 4 depicts the values of the effective multiplication factor for different spatial (spanning the range from one step to the 8-step pin approximation) and angular resolutions, ranging from $2 to $16. The reference eigenvalue of 1,18655, which has been derived from multi-group MCNP calculations, is also shown as dashed line. We shall make a few general remarks on the results from this graph: firstly, one may

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161

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more than two spatial steps show a negative bias compared to the reference calculation, with our bestestimate value for the 8-step/S16-calculation amounting to 1,18512, thus being approximately 0,12% below the reference and slightly better than the value of ke~=1,184818 we have originally submitted. We shall make a few comments on this behavior in section 4. As additional information, we show in Figs. 5 and 6 the maximum and average (RMS) pin power deviation from the reference solution, again for different degrees of spatial and angular approximation. We have completely discarded the curves corresponding to the S2-solution, since it resulted in a maximum deviation of approximately 9% and an RMS-deviation of more than 2% and would thus have disturbed the clarity of the graphs. It should be noted that we find a behavior similar to the eigenvalue from Fig. 4: both maximum and RMS error show the tendency to start flattening very early for spatial approximations as low as only three to four steps per pin cell; this statement even holds for rather low angular quadrature sets like $6. The convergence of the results for the angular variable is also nicely visible, as improving the angular resolution e.g. from $12 to $16 hardly yields any further improvement in the error measures. Our bestestimate values amount (for the 8-step/S16 calculation) to ~0,98% maximum and ~0,29% RMS error. This is again slightly better than the results we originally submitted (1,12% maximum error and 0,34% RMS error) and can obviously be attributed to the improved quality of the TWODANT quadrature sets.

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Fig. 8. The eigenvalue behavior for the Chebychev-Legendre quadrature. In the context of this benchmark, we found it also worthwhile to investigate the validity of diffusion theory for this kind of problem. For this kind of calculation, we have used the finite-difference diffusion module of DORT, which uses a simple diffusion coefficient constructed from 1/3otr (otr taken from the C5G7 library). The corresponding results are shown in Fig. 7, now together with our "best-estimate" S16-solution. It is noticeable that even the diffusion approximation yields a rather reasonable eigenvalue of k~fr=l.18308, which is only 0.3% away from the reference value. The maximum pin power deviation, however, is as large as 5%, with a RMS error of approximately 1,5%. Moreover, these values hardly change, when improving the spatial resolution of the problem; in fact we find that a lower spatial approximation gives results for the pin power distribution, which are even closer to the reference. Finally we explored, how the accuracy of our results would improve, when instead of the level-symmetric quadrature sets the Chebychev-Legendre quadratures are used. We have repeated our calculations with the same variations in the spatial variable and for Discrete Ordinates sets ranging from $2 to $16. Since the

162

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Chebychev-Legendre quadrature also allows for higher orders, we performed single additional calculations for the 8-step spatial approximation employing $20, Sz4 and $30. The numerical effort increases dramatically, since the S30-approximation contains 930 angular directions (which is to be compared to the level-symmetric quadrature S~6-set with only 160 angles). We find a behavior very similar to the results obtained for the level-symmetric quadrature sets. Again, a high-order angular approximation guarantees for a rather fast convergence in the spatial variable. Figure 8 shows the eigenvalue, Figs. 9 and 10 the maximum and the root mean square deviation from the reference solution, respectively.

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As may be noted from Fig. 8, the eigenvalue comes now very close to the reference eigenvalue, particularly for the extreme quadratures $20, $24 and $30. It is also noticeable that the error measures start from slightly larger values for low spatial and angular approximations, but become soon favorable over the level-symmetric results. For a direct comparison, we have summarized the main results in Table IV. The eigenvalue of our most elaborate $30 calculation is less than 2.10 .4 away from the reference eigenvalue, while the root mean square has almost halved in comparison to our originally submitted results. Nevertheless, it should not be forgotten that the numerical effort for these calculations is about an order of magnitude larger than for e.g. a level-symmetric S12-calculation. To illustrate this, we have in Table IV also given the CPU times necessary to converge the solution to better than 5-10 -8 in k~ff and 5.10 v in the nodal fission source. The calculations were performed on one POWER4-processor of an IBM p690 server running at 1,3 GHz (The speed of such a processor is, at least for DORT, roughly equal to a typical PC Intel processor running at 3,0 GHz).

Table IV. A comparison of our results, as obtained for different angular quadrature sets. Calculation Type Reference Solution Submitted Results, 816 Level-symmetric $16 TWODANT-Quadrature Chebychev-Legendre Quadrature $16 Chebychev-Legendre Quadrature $24 Chebychev-Legendre Quadrature $3o

Ei~envalue 1,186550 1,184818 1385125

Maximum Error

RMS error

CPU Time

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4. SUPPLEMENTARY STUDIES ON PIN C E L L F L U X D I S T R I B U T I O N S Although the results given in the last sections are apparently in good agreement with the reference solution, it cannot be overlooked that particularly our eigenvalue results show a systematic bias towards smaller values. Similarly, the error measures for the pin power distribution are very hard to minimize and remain almost constant with increasing spatial and angular resolution. Such a behavior is not yet satisfactory, so we tried to figure out to which effect these systematic deviations could be attributed to. For this purpose, we defined a simple sub-problem of the benchmark, considering an eigenvalue problem for only a single pin cell containing UO2 fuel, having reflected boundary conditions. By refining both spatial grid and angular quadrature order, we found some rather remarkable features in the flux solutions, which gave us at least an idea where the persistent deviations might stem from. As mentioned above, the spatial approximation of the pin cell has obviously far less effect on the quality of the C5G7 calculations than the angular modeling. To confirm this statement, we performed some additional calculations on the pin cell, using a high quadrature order of $32 with varying mesh spacings. In Figs. 11 and 12 we show the behavior of the resulting fast (energy group 1) and thermal (energy group 7) flux profile along the diagonal of the pin cell. The pin has been approximated by 8x8-, 26x26 and 66x66 spatial meshes, as described at length in section 2. It can clearly be seen that even the coarse 8x8-grid represents the flux distribution inside the cell sufficiently well, the deviations at the pin corners being much less than a percent for both fast and thermal flux.

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0,6

0,8

1,0

1,2

o

.,o ~ \,,.,, ,~o ~, po.~

.~~ %~,--,~ 1,4

1,6

1,8

Diagonal Coordinate Fig. 13. The fast flux profile inside a C5G7

U O 2 pin

cell for varying angular resolutions.

0,50 0,48, c~ O .%...,

0,46-

(.9 0,44.

'~i

0~ cELI 0,42-

--u- 36 x 36-Grid, S,

v

\~

36x36-Gnd, S16 .

]¢'

//

.q

m

U.. 0,40 (D ct-0,36

"" '

0,0

I

!

I

1

0,2

0,4

0,6

0,8

!

1,0

•

I

II

1,2

1,4

!

1,6

'1

1,8

Diagonal Coordinate Fig. 14. The thermal flux profile inside a C5G7 UO2 pin cell for varying angular resolutions.

166

A. Pautz et al

Varying the angular quadrature order instead has a much larger effect on the pin cell fluxes. We illustrate this in Figs. 13 and 14, where we compare thermal and fast fluxes for the same problem geometry (i.e. a fixed spatial 18xl8-grid), but gradually increasing quadrature orders, ranging from $4 to Sloo (if not mentioned otherwise, we use the Chebychev-Legendre quadrature sets throughout this section). As for the fast flux (Fig. 13), we encounter major deviations of the order of several percent between the low $4/$8 angular approximation and the most accurate $100 results. This holds both for the interior region representing the fuel rod as well as for the exterior, moderating part of the cell. It is remarkable that even the $32 curve significantly underestimates the fast flux in the center, while it clearly overestimates it at the cell edges. To avoid the rather unphysical oscillations over the extent of the cell and to obtain fluxes accurate to better than one percent, we would thus recommend using SN-orders beyond $40 or so, which would, however, require an enormous (and usually unacceptable) numerical effort. Apparently, the angular quadrature order has a somewhat smaller influence on higher groups, as can be seen from Fig. 14, which shows the corresponding situation for the thermal flux inside the UO2 pin cell. The flux shape is now well reproduced even by low quadrature orders, with only the S4-solution giving visibly larger deviations. However, the absolute error in the fast fluxes from Fig. 13 propagates through the whole energy group structure and gives rise to deviations in the amplitude of the thermal fluxes as well. Although hardly visible in Hg. 14 due to the scaling of the graph, the differences between e.g. $8 and $32 calculation still amount to approximately 0.5-1.0% at some positions inside the fuel. When considering MOX fuel instead (not shown in this paper), these deviations may be as large as 2%. These deviations certainly change the fission distribution as well; since the thermal groups contribute the largest part to the overall fission source, even small flux deviations can have a significant and non-negligible effect. To quantify this effect, we considered the changes in eigenvalue with increasing quadrature order. Since the k~ff-vatue for the pin cell problem is simply the ratio of production to absorption rate, we evaluated these quantities and found that the absorption rate is mainly preserved under change of quadrature order, while the fission production rate increased monotonously with higher SN-orders. It is therefore the fission source contribution, which causes the variations in keff. For a UO2-pin cell, we obtained an eigenvalue change of approximately 0.2% when going from $4 to S100, while for a MOX pin cell this value can be as large as 1%, which can be attributed to the "harder" energy spectrum. It is therefore at least plausible that the flux distortions in the fast group may cause significant deviations in both eigenvalue and pin power distribution. The fact that these deviations are in general larger for MOX pin cells is consistent with our finding that the MOX pin powers in the C5G7 benchmark show larger errors than the uranium oxide pins. It may therefore be a good advice to handle such situations with great care. SN-production codes like DORT and TWODANT e.g. offer the possibility to specify the Discrete Ordinates order in a groupwise manner, such that e.g. high SN-orders would apply to the fastest energy group only, while low SN-orders would suffice for the thermal range. Finally, we shall draw the reader's attention to another interesting effect concerning the choice of the quadrature set. Besides the quadrature order, we have also compared the differences arising from the use of either the Chebychev-Legendre or the standard Gaussian quadrature set. The results are depicted in Figs. 15 and 16, again for both fast and thermal fluxes. Obviously, even the choice of quadrature sets can strongly influence the fast flux shape inside the cell and produces more or less pronounced, unphysical oscillations, which appear to be somewhat stronger for the Gaussian quadrature set. This also explains why we obtain slightly "better" results using the Chebychev-Legendre quadrature instead of the standard set. For completeness, we depict in Fig. 16 the analogous situation for the thermal fluxes: as opposed to the high energy range, we hardly encounter any difference in the flux shape between distinct quadrature sets for thermal energies. To summarize, we have shown in this section that the choice of both quadrature order and quadrature set influences the quality of the fast flux solutions significantly and gives rise to the deviations in eigenvalue and pin power we observed for our C5G7 solutions. As shown above, it may therefore be worthwhile to consider alternative SN-quadrature sets or group-dependent quadrature orders.

167

Discrete ordinates code DORT

1,19]

-

1,18 t:~

~,,L~:L~

~

~;~

.&...-

1, 171 ':;' ~! oI..., (.9 1,16- I

,,5

,~

~ ' ~ ..,'~

' ~'- ~

"~!~1

~'~

1,15 ; o (-*

1,14 ~

'tT',

LI..

---13-- 66 x 66, Chebychev-Legendre - o - 66 x 66, Gausslan Quadrature .~, 66 x 66, Chebychev-Legendre -.~,- 66 x 66, Gaussian Quadrature

U_ 1,12 1,11 0,0

I

I

I

I

0,2

0,4

0,6

0,8

!

I

1,0

kL, -

S~~ # t S8 ~ , : ~ S~e ,::~ .... S~e I

1,2

1,4

I

I

1,6

1,8

DiagonalCoordinate(cm) Fig. 15. The fast flux profile inside a C5G7

pin cell for Chebychev-Legendre and standard Gaussian quadrature set.

UO2

0,50

0,48 i ~

-~ 66*66, Chebychev-LegendreS~6 , ~

i'---

Q..

0

0,46

LO 0,44. cLU m

LI..

c-F-

0,42. 0,400,38 0,36 0,0

I

I

I

I

I

I

0,2

0,4

0,6

0,8

1,0

1,2

I

1,4

!

1,6

I

1,8

DiagonalCoordinate Fig. 16. The thermal flux profile inside a C5G7 UO2 pin cell for Chebychev-Legendre and standard Gaussian quadrature set.

168

A. Pautz et al

5. CONCLUSION To conclude, the results obtained in the C5G7-benchmark reproduce the reference solution from MonteCarlo calculations quite well and show, that even with a rather low spatial resolution of the fuel pin (three steps may already be a sufficient representation) and a medium angular discretization (e.g. Ss) the necessary accuracy can be achieved.

ACKNOWLEDGMENT

The investigations reported in section 4 were essentially stimulated by a close cooperation with Edgar Kiefhaber from Forschungszentrum Karlsruhe (Institute for Nuclear and Energy Technologies), whose support is gratefully acknowledged. REFERENCES

1. Nuclear Energy Agency, "Deterministic Transport Calculations without Spatial Homogenization", NEA/NSC/DOC(2003)16, available from www.nea.fr, (2003). 2. Y.Y. Azmy, "What's New with DOORS", PHYSOR 2000 Workshop, Pittsburgh, May 7-11, (2000). 3. A. Pautz and A. Birkhofer, "DORT-TD: A Transient Neutron Transport Code with Fully Implicit Time Integration", Nuclear Science and Engineering, 145, p. 299-319, (2003). 4. R.E. Alcouffe et al., "DANTSYS: A Diffusion Accelerated Neutral Particle Transport Code System", Los Alamos National Laboratory Report LA-12969-M, (1995).