A New
Runge-Kutta-Fehlberg
Burn-up Solver
Charles A. Wemple
2011 International Users Group Meeting
Stockholm, Sweden | May 24-25, 2011
Overview
• Introduction
• Overview of Burn-up Methods
• Runge-Kutta-Fehlberg Method
• Implementation of R-K-F Method
• Test Results
• Summary, Conclusions, and Future Work
May 24-25, 2011
International Users Group Meeting
Introduction
• Lattice codes have two fundamental functions
– Solve transport equation
– Calculate isotopic transmutation (burn-up)
• Great improvements in solving transport equation
• Few improvements in solving burn-up equation – why?
– Present methods work pretty well
– Existing solver is quite fast – small fraction of total CPU time
– More rigorous methods require more CPU time
• Modern computers limit penalties for higher accuracy
• Conclusion – time to look at burn-up solvers!
May 24-25, 2011
International Users Group Meeting
Burn-up – What are we solving?
May 24-25, 2011
International Users Group Meeting
Burn-up – What are we solving? (cont.)
• Make approximation of “interval invariance”
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Divide time variable into multiple intervals
Assume flux constant over each interval
Then, reaction rates are also constant
Only isotopic concentrations vary during interval
• Reduces integro-differential to ordinary differential
– Simpler to solve
– With predictor-corrector, errors are essentially negligible
May 24-25, 2011
International Users Group Meeting
Burn-up Methods – A Trade-off Study
• Chain Linearization
– Extremely fast
– Limited accuracy
• Chain approximations
• Limited reaction paths
May 24-25, 2011
International Users Group Meeting
Burn-up Methods – A Trade-off Study (cont.)
• Matrix Exponential Method
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Widely used (ORIGEN, etc.)
Well tested – many years of experience
Matrix rapidly gets large – NxN, where N is number of nuclides included
Numerical methods all have limitations - see Moler and VanLoan, SIAM
Review, 45 (1), 3-49
• Runge-Kutta methods
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–
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Widely used for solving differential equations
Great variety of solutions available
Excellent numerical properties
Very accurate (if applied properly)
But…..fairly slow
May 24-25, 2011
International Users Group Meeting
Runge-Kutta-Fehlberg Method
May 24-25, 2011
International Users Group Meeting
Runge-Kutta-Fehlberg Method (cont.)
May 24-25, 2011
International Users Group Meeting
Runge-Kutta-Fehlberg Method (cont.)
• For Runge-Kutta-Fehlberg, solve R-K twice
– First for qth-order equations
– Again for (q+1)th-order equations
– Adjust step size, h, to force difference (local error) below defined level
• Chose 4th-order R-K-F method
• Selected widely-used coefficient set
i
ai
bi0
bi1
bi2
bi3
bi4
0
1
1/4
1/4
2
3/8
3/32
9/32
3
12/13
1932/2197
-7200/2197
7296/2197
4
1
439/216
-8
3680/513
-845/4104
5
1/2
-8/27
2
-3544/2565
1859/4104
May 24-25, 2011
International Users Group Meeting
-11/40
ci
di
25/216
16/135
0
0
1408/2565
6656/12825
2197/4104
28561/56430
-1/5
-9/50
2/55
Implementation of R-K-F Method
May 24-25, 2011
International Users Group Meeting
Implementation of R-K-F Method (cont.)
• Problem – how to define step size, h?
– Iteration in standard R-K-F method is computationally inefficient
– Testing showed solution stability dependent on just a few isotopes
•
•
105Ru
and 105Rh for equilibrium I/Xe
135I and 135Xe for non-equilibrium
– Step sizes longer than ~5x half-life produced instabilities
• Initial solution – fixed step size
– Equilibrium – h=75 000 sec
– Non-equilibrium – h=25 000 sec
– Decay – 32 steps per interval
May 24-25, 2011
International Users Group Meeting
Implementation of R-K-F Method (cont.)
• Problem – still pretty slow
– Inherently dependent on length of burn-up interval
– Acceptable for short intervals; unacceptable for longer intervals
• Testing demonstrated two important characteristics
– Short steps necessary for start of burn-up interval
– Longer steps after concentrations saturate
• Solution – sub-stepping
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Break each burn-up interval into sub-intervals
Very short steps for early sub-intervals; increase as interval progresses
Isotope-dependent sub-interval and step definitions
Sub-intervals and step sizes defined to reproduce fine-step results
• Reduced computation time by ~3x
May 24-25, 2011
International Users Group Meeting
Test Results
• R-K-F method implemented in HELIOS2
• Basic testing - variety of pin-cell calculations
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UO2
MOX
UO2-Gd2O3
U-Th
• Summary of results
– Eigenvalue difference up to ~600pcm
– Small differences in major isotope concentration - ~0.5%
– Large differences in minor isotope concentrations
May 24-25, 2011
International Users Group Meeting
Test Results (cont.)
• Yankee Rowe isotopics benchmark
– 18x18 PWR with control blades
– Westinghouse performed destructive pin analyses (1966 & 1969)
– Isotopics for actinides only – U → Cm
• Summary of results
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Eigenvalue differences less than 200pcm
Small differences in main actinides – up to ~0.3%
Larger differences in lesser actinides – up to several percent
Improved agreement with measured concentrations
May 24-25, 2011
International Users Group Meeting
Summary, Conclusions, and Future Work
• Developed new Runge-Kutta-Fehlberg burn-up solver
• Test implementation in HELIOS2
• Improved isotopic concentration results
• Slower, but more accurate, than linearized chains
• Will be released in HELIOS version 2.1
• May be implemented in CASMO5
May 24-25, 2011
International Users Group Meeting
International Users Group
Meeting
September 2009