NAME OR DESIGNATION OF PROGRAM, COMPUTER, DESCRIPTION OF PROGRAM OR FUNCTION, METHOD OF SOLUTION, RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM, TYPICAL RUNNING TIME, UNUSUAL FEATURES OF THE PROGRAM, RELATED AND AUXILIARY PROGRAMS, STATUS, REFERENCES, MACHINE REQUIREMENTS, LANGUAGE, OPERATING SYSTEM UNDER WHICH PROGRAM IS EXECUTED, OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS, NAME AND ESTABLISHMENT OF AUTHORS, MATERIAL, CATEGORIES

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Program name | Package id | Status | Status date |
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TEMPS | NESC9653/01 | Tested | 29-SEP-1988 |

Machines used:

Package ID | Orig. computer | Test computer |
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NESC9653/01 | CDC CYBER 175 | CDC CYBER 830 |

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3. DESCRIPTION OF PROGRAM OR FUNCTION

TEMPS numerically determines the scalar flux as given by the one-group neutron transport equation with a pulsed source in an infinite medium. Standard plane, point, and line sources are considered as well as a volume source in the negative half-space in plane geometry. The angular distribution of emitted neutrons can either be isotropic or monodirectional (beam) in plane geometry and isotropic in spherical and cylindrical geometry. A general anisotropic scattering Kernel represented in terms of Legendre polynomials can be accommodated with a time- dependent number of secondaries given by c(t)=csub0(t/tsub0)**beta, where beta is greater than -1 and less than infinity. TEMPS is designed to provide the flux to a high degree of accuracy (4-5 digits) for use as a benchmark to which results from other numerical solutions or approximations can be compared.

TEMPS numerically determines the scalar flux as given by the one-group neutron transport equation with a pulsed source in an infinite medium. Standard plane, point, and line sources are considered as well as a volume source in the negative half-space in plane geometry. The angular distribution of emitted neutrons can either be isotropic or monodirectional (beam) in plane geometry and isotropic in spherical and cylindrical geometry. A general anisotropic scattering Kernel represented in terms of Legendre polynomials can be accommodated with a time- dependent number of secondaries given by c(t)=csub0(t/tsub0)**beta, where beta is greater than -1 and less than infinity. TEMPS is designed to provide the flux to a high degree of accuracy (4-5 digits) for use as a benchmark to which results from other numerical solutions or approximations can be compared.

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4. METHOD OF SOLUTION

A semi-analytic method of solution is followed. The main feature of this approach is that no discretization of the transport or scattering operators is employed. The numerical solution involves the evaluation of an analytical representation of the solution by standard numerical techniques. The transport equation is first reformulated in terms of multiple collisions with the flux represented by an infinite series of collisional components. Each component is then represented by an orthogonal Legendre series expansion in the variable x/t where the distance x and time t are measured in terms of mean free path and mean free time, respectively. The moments in the Legendre reconstruction are found from an algebraic recursion relation obtained from Legendre expansion in the direction variable mu. The multiple collision series is evaluated first to a prescribed relative error determined by the number of digits desired in the scalar flux. If the Legendre series fails to converge in the plane or point source case, an accelerative transformation, based on removing the singular portion of the flux (near the wave front) is applied. A scattering kernel is supplied to test the anisotropic scattering option. In addition, fission is accomodated in the isotropic scattering component.

A semi-analytic method of solution is followed. The main feature of this approach is that no discretization of the transport or scattering operators is employed. The numerical solution involves the evaluation of an analytical representation of the solution by standard numerical techniques. The transport equation is first reformulated in terms of multiple collisions with the flux represented by an infinite series of collisional components. Each component is then represented by an orthogonal Legendre series expansion in the variable x/t where the distance x and time t are measured in terms of mean free path and mean free time, respectively. The moments in the Legendre reconstruction are found from an algebraic recursion relation obtained from Legendre expansion in the direction variable mu. The multiple collision series is evaluated first to a prescribed relative error determined by the number of digits desired in the scalar flux. If the Legendre series fails to converge in the plane or point source case, an accelerative transformation, based on removing the singular portion of the flux (near the wave front) is applied. A scattering kernel is supplied to test the anisotropic scattering option. In addition, fission is accomodated in the isotropic scattering component.

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5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM

Maxima of -

140 terms in the Legendre series expansion

100 terms in the multiple collision expansion of moments

4 passes for convergence

The number of desired flux calculations, anisotropy of the scattering kernel, and computational accuracy are limited only by the computer storage available through the use of dynamic storage allocation. The largest time t allowed is limited by the greatest floating point number allowed. For x/t = 1 or -1 and a beam source, the Legendre series may not converge to the accuracy desired as a result of its poor representation at discontinuities.

Maxima of -

140 terms in the Legendre series expansion

100 terms in the multiple collision expansion of moments

4 passes for convergence

The number of desired flux calculations, anisotropy of the scattering kernel, and computational accuracy are limited only by the computer storage available through the use of dynamic storage allocation. The largest time t allowed is limited by the greatest floating point number allowed. For x/t = 1 or -1 and a beam source, the Legendre series may not converge to the accuracy desired as a result of its poor representation at discontinuities.

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6. TYPICAL RUNNING TIME

The computational time required is problem dependent and therefore cannot be easily specified. For a case where the flux is desired to four place accuracy at 23 time points and 5 space points for an isotropic scattering kernel, the computational time is less than 2.5 seconds on a CDC CYBER175 or about 0.012 second per time and space point.

The computational time required is problem dependent and therefore cannot be easily specified. For a case where the flux is desired to four place accuracy at 23 time points and 5 space points for an isotropic scattering kernel, the computational time is less than 2.5 seconds on a CDC CYBER175 or about 0.012 second per time and space point.

NESC9653/01

The three test cases included in this package have been run by NEA-DB on a CDC CYBER 830 computer in less than 2 minutes of CPU time.[ top ]

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10. REFERENCES

- B.D. Ganapol

Solution of the One-Group Time Dependent Neutron Transport

Equation in an Infinite Medium by Polynomial Reconstruction,

Proceedings of the International Meeting on Advances in Nuclear

Engineering Computational Methods,

Knoxville, Tennessee, April 9-11, 1985, Vol. 2, pp. 696-707, 1985, also available in Nuclear Science and Engineering, Vol. 92, pp.

272-279, 1986.

- B.D. Ganapol

Solution of the One-Group Time Dependent Neutron Transport

Equation in an Infinite Medium by Polynomial Reconstruction,

Proceedings of the International Meeting on Advances in Nuclear

Engineering Computational Methods,

Knoxville, Tennessee, April 9-11, 1985, Vol. 2, pp. 696-707, 1985, also available in Nuclear Science and Engineering, Vol. 92, pp.

272-279, 1986.

NESC9653/01, included references:

- B.D. Ganapol:TEMPS: The Evaluation of the One-Group Time Dependent Neutron

Transport Flux in Infinite Media.

University of Arizona Engineering Experiment Station Report (1985)

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11. MACHINE REQUIREMENTS

3 FORTRAN units in addition to input/output units, 20 and 21, respectively.

3 FORTRAN units in addition to input/output units, 20 and 21, respectively.

NESC9653/01

138,000 (octal) words of CDC CYBER 830 main storage.[ top ]

NESC9653/01

NOS 2.5.1 (CDC CYBER 830).[ top ]

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NESC9653/01

File name | File description | Records |
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NESC9653_01.001 | Information file | 57 |

NESC9653_01.002 | JCL and control information | 14 |

NESC9653_01.003 | TEMPS FORTRAN source | 1333 |

NESC9653_01.004 | Sample problem input 1 | 15 |

NESC9653_01.005 | Sample problem input 2 | 14 |

NESC9653_01.006 | Sample problem input 3 | 14 |

NESC9653_01.007 | Sample problem output 1 | 169 |

NESC9653_01.008 | Sample problem output 2 | 2674 |

NESC9653_01.009 | Sample problem output 3 | 73 |

Keywords: legendre polynomials, one-group theory, series expansion, time dependence.