Introduction of the Adding and Doubling Method for Solving Bateman Equations for Nuclear Fuel Depletion

Abstract

This paper introduces and evaluates the Adding and Doubling Method (ADM) for solving the Bateman equations for depletion systems with varying numbers of nuclides and compares it to the Chebyshev Rational Approximation Method (CRAM), both implemented in the reactor physics analysis application Griffin. ADM, when applied to the Crank-Nicolson Finite Difference method, can produce results comparable in accuracy and precision to CRAM with comparable run times for systems with 35 or 297 nuclides. For systems with more than 300 nuclides, the matrix-matrix operations required by ADM are significantly more costly than the matrix-vector operations required by CRAM, making CRAM the more efficient method for systems with large numbers of nuclides. ADM is an accurate method that maintains other advantages over CRAM in that it does not depend on pre-generated coefficients or require complex number operations. ADM also manages to outperform CRAM by a factor of more than 250 in terms of run time for depletion systems that require multiple Bateman solves while the depletion matrix and time step size remain constant over all depletion intervals.

Keywords:

• Adding and Doubling Method
• Bateman equation
• depletion
• Chebyshev Rational Approximation Method

Acronyms

ADM	=	Adding and Doubling Method
ANARD	=	average nuclide absolute relative difference
BOL	=	beginning of life
CNFD	=	Crank-Nicolson Finite Difference
CRAM	=	Chebyshev Rational Approximation Method
DMI	=	direct matrix inversion
ENDF	=	Evaluated Nuclear Data File
INL	=	Idaho National Laboratory
IPF	=	incomplete partial fraction
MNARD	=	maximum nuclide absolute relative difference
NNND	=	negative nuclide number density
OECD	=	Organisation for Economic Co-operation and Development
ORIGEN	=	Oak Ridge Isotope Generation and Depletion Code
ORNL	=	Oak Ridge National Laboratory
PFD	=	partial fraction decomposition
PWR	=	pressurized water reactor
SFR	=	Sodium Fast Reactor
SGE	=	sparse Gaussian elimination

Acknowledgments

We would like to thank Javier Ortensi and Yaqi Wang for their support in implementing this functionality in Griffin. The second author would graciously like to thank INL for hosting his summer visits during completion of this work.

DISCLOSURE STATEMENT

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was funded at INL under the Nuclear Energy Advanced Modeling and Simulation program managed by the U.S. Department of Energy Office of Nuclear Energy, under the U.S. Department of Energy Idaho Operations Office contract DE-AC07-05ID14517.