last modified: 06-OCT-2003 | catalog | categories | new | search |

CCC-0276 DOT-3.5.

GRUNCLE, 1st Collision Source Calculation for Program DOT
DOT-3.5, 2-D Neutron Transport, Gamma Transport Program DOT with New Space-Scaling

top ]
1. NAME OR DESIGNATION OF PROGRAM:  DOT-3.5.
top ]
2. COMPUTERS
To submit a request, click below on the link of the version you wish to order. Only liaison officers are authorised to submit online requests. Rules for requesters are available here.
Program name Package id Status Status date
DGRAD/TPERT CCC-0276/01 Tested 01-SEP-1980
DOT-3.5 CCC-0276/02 Tested 01-MAY-1979
GRTUNCL CCC-0276/03 Tested 01-DEC-1976
VIP CCC-0276/04 Tested 01-MAR-1978
DOT-3.5/E CCC-0276/07 Tested 01-MAY-1979
DUCT CCC-0276/09 Tested 01-DEC-1976

Machines used:

Package ID Orig. computer Test computer
CCC-0276/01 IBM 360 series IBM 370 series
CCC-0276/02 IBM 370 series IBM 370 series
CCC-0276/03 IBM 370 series IBM 370 series
CCC-0276/04 IBM 370 series IBM 370 series
CCC-0276/07 IBM 370 series IBM 370 series
CCC-0276/09 IBM 370 series IBM 370 series
top ]
3. NATURE OF PROBLEM OR FUNCTION

DOT solves the Boltzmann transport equation in two-dimensional geometries. Principal applications are to neutron and/or photon transport, although the code can be applied to transport problems for any particles not subject to external force fields. Both homogeneous andexternal-source problems can be solved. Searches on multiplication factor, time absorption, nuclide  concentration, and zone thickness are available for reactor problems. Numerous edits and output data sets for subsequent use are available.

     DOT-3.5 improves the space-scaling algorithm.

     DOT-3.5/CAB contains group by group UPSCATTER scaling method.

DUCT calculates perturbations to the scalar flux caused by the presence of ducts filled with coolant.

VIP is a program for cross section sensitivity analysis using two- dimensional discrete ordinates transport calculations.

DGRAD calculates the directional flux gradients from DOT-3 diffusion theory flux tapes. In conjunction with VIP and TPERT, it allows the  use of diffusion theory fluxes to obtain exact and first-order perturbation reactivity changes. In order to calculate the reactivity associated with changes in reactor compositions using diffusion theory, it is necessary to fold not only the scalar fluxes with the appropriate cross sections, but also the average flux gradients with the diffusion coefficients. Since DOT diffusion theory does not directly calculate these gradients, it was necessary to calculate the needed quantities external to the DOT code.

TPERT is a perturbation code to obtain exact and first-order reactivity changes. TPERT is coupled to VIP which generates adjoint  forward fux tables using DOT-3 scalar flux tape information.

GRTUNCL calculates an analytical first-collision source for subsequent use in DOT.
top ]
4. METHOD OF SOLUTION

The method of discrete ordinates is used.
Balance equations are solved for the density of particles moving along discrete directions in each cell of a two-dimensional spatial  mesh. Anisotropic scattering is treated using a Legendre expansion of arbitrary order. Convergence can be accelerated by several optional schemes, including a pointwise rescaling technique.

DOT-3.5/E: Differs from DOT-3.5 in that exponential supplementary equations, as well as the usual diamond and weighted schemes, may be used to find the mesh-centre flux from the fluxes at the faces of the mesh.
The model:
1. always gives positive solutions and does not require any fixup techniques provided that the source is non-negative;
2. improves convergence rate in most neutron deep-penetration problems and, for any practical spatial discretization, always requires CPU times not only smaller than those required by DOT-3 mixed (linear + step fixup) model, but also shorter (generally 10-20%) than the times required by DOT-3.5 weighted difference model;
3. increasing spatial mesh size supplies solutions which are always  reasonable overestimates of the exact solution and its numerical behaviour is more stable and coherent than the mixed mode.
Experience up to now from several deep penetration problems in (r,z) and (x,y) geometry shows that, while for neutrons the exponential model almost always works very well, for gamma rays its behaviour may be critical and in some cases there is lack of convergence.
top ]
5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM

The total storage requirement is determined by a formula given in the input description. No other restrictions to problem size are applicable.
top ]
6. TYPICAL RUNNING TIME

Problems of practical value may range from 0.05 minutes to many hours in running time. A formula given in the input description allows estimation of central processor time for a  particular problem.
DOT-3.5/E: In the most deep penetration problems, DOT-3.5/E with exponential model is about 10 to 20 % faster than DOT-3.5 with weighted difference model.
top ]
7. UNUSUAL FEATURES OF THE PROGRAM

Flexible dimensioning is used throughout so that only the total storage requirement for a given problem is of concern. The program has two levels of external storage utilization in order to adapt to various problem requirements efficiently. Cross sections can be input on cards, from an ANISN-type nuclide-organized data set, or from a special group-organized data set which is valuable for very large cross section sets. A free-field data input format facilitates problem preparation. Special features which make DOT-3 well adapted to shielding problems include albedo boundary conditions, the capability to output fluxes at exterior or interior boundaries for use in 'bootstrap' problems, and an analytic first-collision source  calculation that mitigates ray-streaming effects in certain problems. Upscattering is allowed, and special routines are able to  synthesize two-dimensional flux guesses from one-dimensional data produced by the ANISN code. Diffusion theory is also available. The  convergence test can be restricted to specified zones.

DOT-3.5/E: Exponential equations to compute mesh centre fluxes.
top ]
8. RELATED AND AUXILIARY PROGRAMS

ANISN - produces one-dimensional flux guesses.
SPACETRAN - calculates the neutron or gamma flux at arbitrary points in space exterior to a two-dimensional right circular cylindrical system which as been calculated by the discrete-ordinates code DOT.
DOQDP: A computer program to generate level symmetric quadrature sets.
ADOQ: combines asymmetric quadrature data with symmetric quadrature data to form "biased" quadrature sets.
ESTOQ: a program to compute first-collision sources for eliminating ray effects in two-dimensional Sn calculations.
DASH: a void tracing and Sn Monte Carlo bridging code. Using angular angular fluxes from cylindrical geometry calculation, it calculates angular fluxes on arbitrary cylindrical geometry surfaces outisde the DOT boundaries. It is useful to reduce ray effects.
DOMINO: a general purpose code for coupling discrete ordinates and Monte Carlo radiation transport calculations.
APSAI: a computer code for plotting fluxes and absorption densities generated by the ANISN code.
top ]
9. STATUS
Package ID Status date Status
CCC-0276/01 01-SEP-1980 Tested at NEADB
CCC-0276/02 01-MAY-1979 Tested at NEADB
CCC-0276/03 01-DEC-1976 Tested at NEADB
CCC-0276/04 01-MAR-1978 Tested at NEADB
CCC-0276/07 01-MAY-1979 Tested at NEADB
CCC-0276/09 01-DEC-1976 Tested at NEADB
top ]
10. REFERENCES

- W. W. Engle, Jr.:
  'A User's Manual for ANISN'
  Computing Technology Center, Union Carbide Corporation
  Report K-1693 (March 30, 1967).
- N.M. Greene, and C.W. Craven, Jr.:
  'XSDRN, A Discrete Ordinates Spectral Averaging Code'
  ORNL-TM-2500 (July 1969).
- W.A. Rhoades:
  'The ALC1 Program for Cross-Section Library Management'
  ORNL-TM-4015 (December 1972).
- Unpublished Notes for GIP, GRTUNCLE, SPACETRAN, FALSTF, ISOPLOT, PLOTTER,
  PERT-2D, FACT, and DOMINO.
- R. Douglas O'Dell and Raymond E. Alcouffe:
  Transport Calculations for Nuclear Analyses: Theory and Guidelines for
  Effective Use of Transport Codes
  LA-10983-MS and UC-32 (September 1987).
- NEA-CPL:
  Note on DOT3.5 (12 January 1977)
- OECD/NEA:
  Note from the NEA Data Bank (May 1984)
- L. Erradi:
  Corrections Apportees a DOT-3.5/E - Adaptation sur la Machine IBM 4361 (4341)
  VM.
CCC-0276/01, included references:
- E.T. Tomlinson and R.A. Lillie:
User's Guide and Description of the Perturbation Code Modules DGRAD and TPERT
ORNL/CSD/TM-71 (September 1978)
- E.T. Tomlinson, R.L. Childs and R.A. Lillie:
DOS Perturbation Module DGRAD/VIP/TPERT, ORNL/CSD/TM-116 (May 1980)
CCC-0276/02, included references:
- W.A. Rhoades:
Comments on the DOT 3.5 Version of DOT III (November 1975)
- Utility Network of America:
Program Additions to DOT 3 (July 1975)
- W.A. Rhoades and F.R. Mynatt:
The DOT-III Two-Dimensional Discrete Ordinates Transport Code
ORNL-TM-4280 (September 1973)
- P.A. Read, W.E. Selph and R.J. Cerbone:
DUCT Code Manual, Gulf-RT-10654
- J.T. West:
SORREL (November 1975)
- J.P. Jenal:
Common Symmetric Quadratures and the DOQDP Computer Code (August 1975)
- R.L. Childs:
VIP Input Instructions (November 1976)
- R.L. Childs, D.E. Bartine and W.W. Engle, Jr.:
Perturbation Theory and Sensitivity Analysis for Two-Dimensional Shielding
Calculations, ANS Transactions, Vol. 21, pp. 543-544 (June 1975)
- J.P. Jenal et al.:
The Generation of a Computer Library for Discrete Ordinates Quadrature Sets
ORNL/TM-6023 (September 1977)
- E.T. Tomlinson and R.A. Lillie:
User's Guide and Description of the Perturbation Code Modules DGRAD and TPERT
ORNL/CSD/TM-71 (September 1978)
- E.T. Tomlinson, R.L. Childs, and R.A. Lillie:
DOS Perturbation Modules DGRAD/VIP/TPERT, ORNL/CSD/TM-116 (May 1980)
- J. Pena:
DOT 3.5/E-JEN, Two Dimensional Discrete Ordinates Radiation Transport Code
- P. Barbucci and F. Di Pasquantonio:
Implementation of Exponential Supplementary Equations on DOT-III and DOT 3.5
Codes (Document received November 1977)
- P. Barbucci and F. Di Pasquantonio:
Exponential Supplementary Equations for Sn Methods: The Two-Dimensional Case
Reprint of Paper, Proceedings of Fifth International Conference on Reactor
Shielding, Knoxville, Tennessee (April 18-22, 1977)
CCC-0276/07, included references:
- W.A. Rhoades:
Comments on the DOT 3.5 Version of DOT III (November 1975)
- Utility Network of America:
Program Additions to DOT 3 (July 1975)
- W.A. Rhoades and F.R. Mynatt:
The DOT-III Two-Dimensional Discrete Ordinates Transport Code
ORNL-TM-4280 (September 1973)
- P.A. Read, W.E. Selph and R.J. Cerbone:
DUCT Code Manual, Gulf-RT-10654
- J.T. West:
SORREL (November 1975)
- J.P. Jenal:
Common Symmetric Quadratures and the DOQDP Computer Code (August 1975)
- R.L. Childs:
VIP Input Instructions (November 1976)
- R.L. Childs, D.E. Bartine and W.W. Engle, Jr.:
Perturbation Theory and Sensitivity Analysis for Two-Dimensional Shielding
Calculations, ANS Transactions, Vol. 21, pp. 543-544 (June 1975)
- J.P. Jenal et al.:
The Generation of a Computer Library for Discrete Ordinates Quadrature Sets
ORNL/TM-6023 (September 1977)
- E.T. Tomlinson and R.A. Lillie:
User's Guide and Description of the Perturbation Code Modules DGRAD and TPERT
ORNL/CSD/TM-71 (September 1978)
- E.T. Tomlinson, R.L. Childs, and R.A. Lillie:
DOS Perturbation Modules DGRAD/VIP/TPERT, ORNL/CSD/TM-116 (May 1980)
- J. Pena:
DOT 3.5/E-JEN, Two Dimensional Discrete Ordinates Radiation Transport Code
- P. Barbucci and F. Di Pasquantonio:
Implementation of Exponential Supplementary Equations on DOT-III and DOT 3.5
Codes (Document received November 1977)
- P. Barbucci and F. Di Pasquantonio:
Exponential Supplementary Equations for Sn Methods: The Two-Dimensional Case
Reprint of Paper, Proceedings of Fifth International Conference on Reactor
Shielding, Knoxville, Tennessee (April 18-22, 1977)
top ]
11. MACHINE REQUIREMENTS

Card input, printed and punched output, and the 3 scratch data sets may be located on any external storage device, as may the 9 optional data sets. A small, but useful, problem can be run in as little as 256k bytes of fast memory. A clock, if available, provides timing data.
top ]
12. PROGRAMMING LANGUAGE(S) USED
Package ID Computer language
CCC-0276/01 FORTRAN-IV
CCC-0276/02 FORTRAN+ASSEMBLER
CCC-0276/03 FORTRAN+ASSEMBLER
CCC-0276/04 FORTRAN-IV
CCC-0276/07 FORTRAN-IV
CCC-0276/09 FORTRAN-IV
top ]
13. OPERATING SYSTEM OR MONITOR UNDER WHICH PROGRAM IS EXECUTED:  IBM
360 OS version 16 or later.
top ]
14. ANY OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS

This is an improved version of DOT-3 and replaces all previous versions of this code.
top ]
15. NAME AND ESTABLISHMENT OF AUTHOR

The DOT code has been developed by Neutron Physics Division
Oak Ridge National Laboratory
Oak Ridge, Tennessee, U.S.A
top ]
16. MATERIAL AVAILABLE
CCC-0276/01
File name File description Records
CCC0276_01.001 JCL FOR COMPILE&LINK TPERT,DGRAD,LIBGEN 90
CCC0276_01.002 DEFILE ASSEMBLER ROUTINE 156
CCC0276_01.003 TPERT SOURCE F4 938
CCC0276_01.004 DGRAD SOURCE F4 700
CCC0276_01.005 LIBGEN SOURCE F4 547
CCC0276_01.006 DOT3.5 INPUT,TRANSPORT FORWARD 59
CCC0276_01.007 DOT3.5 INPUT,TRANSPORT ADJOINT 59
CCC0276_01.008 XSEC DATA FOR LIBGEN (TRANSPORT) 48
CCC0276_01.009 VIP INPUT DATA 7
CCC0276_01.010 TPERT INPUT, TRANSPORT 22
CCC0276_01.011 DOT3.5 OUTPUT, TRANSPORT FORWARD 1727
CCC0276_01.012 DOT3.5 OUTPUT, TRANSPORT ADJOINT 1804
CCC0276_01.013 VIP PRINTED OUTPUT, TRANSPORT 208
CCC0276_01.014 LIBGEN PRINTED OUTPUT, TRANSPORT 149
CCC0276_01.015 TPERT OUTPUT, TRANSPORT 194
CCC0276_01.016 JCL FOR RUNNING DOT3.5,VIP,LIBGEN,TPERT 104
CCC0276_01.017 X-SEC DATA FOR LIBGEN (DIFFUSION) 16
CCC0276_01.018 DOT3.5 INPUT, DIFFUSION FORWARD 47
CCC0276_01.019 DGRAD INPUT, FORWARD 7
CCC0276_01.020 DOT3.5 INPUT, DIFFUSION ADJOINT 47
CCC0276_01.021 DGRAD INPUT, ADJOINT 7
CCC0276_01.022 TPERT INPUT, DIFFUSION 22
CCC0276_01.023 LIBGEN OUTPUT, DIFFUSION 41
CCC0276_01.024 DOT3.5 OUTPUT, DIFFUSION FORWARD 1296
CCC0276_01.025 DGRAD OUTPUT, FORWARD 1029
CCC0276_01.026 DOT3.5 OUTPUT, DIFFUSION ADJOINT 1333
CCC0276_01.027 DGRAD OUTPUT, ADJOINT 1029
CCC0276_01.028 VIP PRINTED OUTPUT, DIFFUSION 208
CCC0276_01.029 TPERT OUTPUT, DIFFUSION 194
CCC0276_01.030 JCL TO RUN LIBGEN,DOT3.5,DGRAD,VIP,TPERT 133
CCC-0276/02
File name File description Records
CCC0276_02.002 DESCRIPTION OF VARIABLES IN COMMON 399
CCC0276_02.003 SOURCE PROGRAM (ORNL FEB-77,F4,EBCDIC) 5946
CCC0276_02.004 HEADER,MESAGE,LOCO,SUBR (F4,EBCDIC) 292
CCC0276_02.005 CNNP SUBROUTINE (FEB-79,F4,EBCDIC) 209
CCC0276_02.006 GRIND SUBROUTINE (F4,EBCDIC) 413
CCC0276_02.007 WWESOL SUBROUTINE (F4,EBCDIC) 124
CCC0276_02.008 GRIND SUBROUTINE (ASSEMBLER,EBCDIC) 1520
CCC0276_02.009 WWESOL SUBROUTINE (ASSEMBLER,EBCDIC) 400
CCC0276_02.010 ALOCAT SUBROUTINE (ASSEMBLER,EBCDIC) 44
CCC0276_02.011 CMVBT SUBROUTINE (ASSEMBLER,EBCDIC) 37
CCC0276_02.012 JOBNUM,ITIME,IOR,IAND,ICLOCK (ASSEMBLER) 236
CCC0276_02.013 CMVBT,ALOCAT,ITIME,IFTIME (F4,EBCDIC) 38
CCC0276_02.014 ERROR MESSAGES 50
CCC0276_02.015 OVERLAY CARDS 27
CCC0276_02.016 SAMPLE PROBLEM INPUT DATA 1027
CCC0276_02.017 SAMPLE PROBLEM PRINTED OUTPUT 7932
CCC0276_02.018 JCL FOR RUNNING DOT-3.5 SAMPLE PROBLEM 193
CCC0276_02.019 UPSCATTER SAMPLE PROB. PRINTED OUTPUT 2678
CCC0276_02.020 VIP SOURCE PROGRAM (F4,EBCDIC) 1086
CCC0276_02.021 BSAM ROUTINES (ASSEMBLER,EBCDIC) 421
CCC0276_02.022 BSAM ROUTINES (F4,EBCDIC) 269
CCC0276_02.023 'FORWARD' FOR VIP SAMPLE PROB. INPUT DATA 386
CCC0276_02.024 'FORWARD' FOR VIP SAMPLE PROB. OUTPUT 3649
CCC0276_02.025 'ADJOINT' FOR VIP SAMPLE PROB. INPUT DATA 381
CCC0276_02.026 'ADJOINT' FOR VIP SAMPLE PROB. OUTPUT 3296
CCC0276_02.027 VIP SAMPLE PROBLEM INPUT DATA 19
CCC0276_02.028 VIP SAMPLE PROBLEM PRINTED OUTPUT 899
CCC0276_02.029 SWANLAKE S. PROB. INPUT(USING VIP OUTPUT) 42
CCC0276_02.030 SWANLAKE S. PROB. PRINTED OUTPUT 304
CCC0276_02.031 JCL FOR DOT-VIP-SWANLAKE RUN 75
CCC0276_02.032 SORREL SOURCE PROGRAM (F4,EBCDIC) 1210
CCC0276_02.033 JCL FOR SORREL SAMPLE PROBLEM 32
CCC0276_02.034 SORREL SAMPLE PROBLEM INPUT DATA 221
CCC0276_02.035 SORREL SAMPLE PROBLEM PRINTED OUTPUT 3729
CCC-0276/03
File name File description Records
CCC0276_03.001 DIRECTOR (COMMENT CARDS) DOT-3 499
CCC0276_03.002 GRIND SOURCE - ASSEMBLER DOT-3 1676
CCC0276_03.003 ICLOCK SOURCE - ASSEMBLER DOT-3 92
CCC0276_03.004 ITIME SOURCE - ASSEMBLER DOT-3 72
CCC0276_03.005 BSAM SOURCE - ASSEMBLER DOT-3 298
CCC0276_03.006 DOT-3 SOURCE (6/74) - FORTRAN IV EBCDIC 6489
CCC0276_03.007 IN-LINE PROCEDURE FOR LOADING AND EXECUTION 52
CCC0276_03.008 JCL FOR GENERATING DOT-3 LOAD MODULE 15
CCC0276_03.009 DOT-3 OVERLAY STRUCTURE 24
CCC0276_03.010 JCL FOR GENERATING DOT-3 LOAD MODULE FOR GRT 9
CCC0276_03.011 DOT-3 CALLING PROGRAM FOR GRTUNCL RUN 7
CCC0276_03.012 GRTUNCL SOURCE (6/74) - FORTRAN IV EBCDIC 1666
CCC0276_03.013 JCL FOR GRTUNCL-DOT3 SAMPLE PROBLEM 38
CCC0276_03.014 DATA FOR GRTUNCL SAMPLE PROBLEM 33
CCC0276_03.015 OUTPUT OF GRTUNCL SAMPLE PROBLEM 603
CCC0276_03.016 DATA FOT DOT-3 SAMPLE WITH GRTUNCL TAPE 43
CCC0276_03.017 OUTPUT OF DOT-3 SAMPLE WITH GRTUNCL TAPE 3105
CCC-0276/04
File name File description Records
VIP SEND DOT-3.5 (CCC-0178/05) 0
CCC-0276/07
File name File description Records
CCC0276_07.002 DESCRIPTION OF VARIABLES IN COMMON 399
CCC0276_07.003 SOURCE PROGRAM (ORNL FEB-77,F4,EBCDIC) 5946
CCC0276_07.004 HEADER,MESAGE,LOCO (ENEL VERSION,F4,EBCDIC) 312
CCC0276_07.005 CNNP SUBROUTINE (FEB-79,F4,EBCDIC) 209
CCC0276_07.006 GRIND SUBROUTINE (ENEL VERSION,F4,EBCDIC) 466
CCC0276_07.007 WWESOL SUBROUTINE (F4,EBCDIC) 124
CCC0276_07.008 WWESOL SUBROUTINE (ASSEMBLER,EBCDIC) 400
CCC0276_07.009 ALOCAT SUBROUTINE (ASSEMBLER,EBCDIC) 44
CCC0276_07.010 CMVBT SUBROUTINE (ASSEMBLER,EBCDIC) 37
CCC0276_07.011 JOBNUM,ITIME,IOR,IAND,ICLOCK (ASSEMBLER) 236
CCC0276_07.012 CMVBT,ALOCAT,ITIME,IFTIME (F4,EBCDIC) 38
CCC0276_07.013 ERROR MESSAGES 50
CCC0276_07.014 OVERLAY CARDS 27
CCC0276_07.015 SAMPLE PROBLEM INPUT DATA 167
CCC0276_07.016 SAMPLE PROBLEM PRINTED OUTPUT 3299
CCC0276_07.017 JCL FOR RUNNING THE SAMPLE PROBLEM 151
CCC0276_07.018 3-GROUP UPSCATTER PRINTED OUTPUT 1202
CCC-0276/09
File name File description Records
CCC0276_09.001 DIRECTOR (COMMENT CARDS) DOT-3 499
CCC0276_09.002 GRIND SOURCE - ASSEMBLER DOT-3 1676
CCC0276_09.003 ICLOCK SOURCE - ASSEMBLER DOT-3 92
CCC0276_09.004 ITIME SOURCE - ASSEMBLER DOT-3 72
CCC0276_09.005 BSAM SOURCE - ASSEMBLER DOT-3 298
CCC0276_09.006 DOT-3 SOURCE (6/74) - FORTRAN IV EBCDIC 6489
CCC0276_09.007 IN-LINE PROCEDURE FOR LOADING AND EXECUTION 52
CCC0276_09.008 JCL FOR GENERATING DOT-3 LOAD MODULE 15
CCC0276_09.009 DOT-3 OVERLAY STRUCTURE 24
CCC0276_09.010 DUCT SOURCE (8/74) - FORTRAN IV EBCDIC 2271
CCC0276_09.011 JCL FOR DOT-3 DUCT SAMPLE PROBLEM 22
CCC0276_09.012 DUCT OVERLAY STRUCTURE 12
CCC0276_09.013 DATA FOR DOT-3 SAMPLE PROBLEM (DUCT CASE) 116
CCC0276_09.014 OUTPUT OF DOT-3 SAMPLE PROBLEM (DUCT CASE) 990
CCC0276_09.015 DATA FOR DUCT SAMPLE PROBLEM USING DOT-3 FLU 276
CCC0276_09.016 OUTPUT OF DUCT SAMPLE PROBLEM USING DOT-3 FL 528
top ]
17. CATEGORIES
  • C. Static Design Studies
  • J. Gamma Heating and Shield Design

Keywords: anisotropic scattering, discrete ordinate method, legendre polynomials, neutron transport theory, two-dimensional.