WPNCS Expert Group on Source Convergence for Criticality Safety Analyses (SCCSA)


Improper source convergence can lead to non-conservative estimates of the k-effective for various fissionable configurations. To improve the robustness of criticality safety analyses with respect to source convergence, in 1999 the NEA established an expert group to investigate the long-standing problem of source convergence for certain classes of nuclear criticality safety problems.

Under the guidance of the Working Party on Nuclear Criticality Safety, the major assignments of the expert group were:

  • developing criticality safety benchmark problems which exhibit convergence problems;
  • testing fission source algorithms for vulnerability to slow convergence;
  • developing criteria to measure convergence reliability;
  • developing source convergence guidelines for the nuclear criticality safety analysts;
  • exploring and evaluating methods to detect source convergence;
  • publishing the results.

The Expert Group on Source Convergence ended activities in 2009.


Elliott Whitesides of ORNL was one of the early advocates of addressing fission source convergence problems, during an era when computing power and memory were severely limited. Those constraints have since been relaxed by orders of magnitude, so we now can and should apply far more computing power to source and statistical convergence and bias reduction. Nevertheless, the same vigilance is necessary to avoid non-conservative results using Monte Carlo Criticality codes. The tools described in the Phase II section below should not be viewed as guarantors of source convergence. Rather, they are tools to help the analyst understand the physical properties of the system and the mathematical properties of the transport equation as applied in his analysis.

The relevant methodological issues arising in Monte Carlo calculations are:

  • Sufficient evolution of the fission source distribution to remove contamination of the higher eigenmodes.
  • Sufficient histories per cycle to eliminate bias and undersampling of important fissile regions.
  • Adequate accounting or mitigation of uncertainty bias due to autocorrelation.

Phase I: Test Cases

Phase I consisted of collection of four test cases, analysis using Monte Carlo codes, and comparison of results, as documented in the following report:

Source Convergence in Criticality Safety Analyses. Phase I: Results for Four Test Problems
OECD NEA Report No. 5346, ISBN 92-64-02304-6 (2006).

Each case embodied difficulties experienced by Monte Carlo criticality calculations, all related to the fission source iteration process. They are not limited to slow or false convergence, but also include undersampling and autocorrelation. The analyst is invited to use these to explore the properties of these systems from a Monte Carlo methods point of view, especially relating to the source iteration method used in his code.

The test case specifications are as follows:

Phase II: Guidance

Phase II of the Expert Group's program consisted of developing guidance for criticality safety analysts to prevent source convergence difficulties from contributing to errors in safety analyses. As described in the Phase II report (below), these fall into three main categories:

  • Convergence tests
  • Convergence acceleration methods
  • Computational strategies

Source Convergence in Criticality Safety Analyses, Expert Group on Source Converegence Phase II Report to OECD/NEA WPNCS, 2011.

There are many references on this subject. A sample of the more recent (and more effective) ones are listed here.

  1. Fundamentals of Monte Carlo Transport, a Monte Carlo course by Forrest B. Brown (LA-UR-05-4983) Includes an introductory description of the wide range of Monte Carlo transport methods, as implemented in MCNP. The section on Eigenvalue calculations begins in Lecture 7, a bit over halfway through. The section includes a bibliography of stationarity tests. A second section on eigenvalue calculations discusses the power iteration, Wielandt acceleration, and the superhistory method implementations and a bibliography.

  2. On the Use of Shannon Entropy of the Fission Distribution for Assessing Convergence of Monte Carlo Criticality Calculations, by Forrest B. Brown, PHYSOR-2006, ANS Topical Meeting on Reactor Physics, Vancouver, Canada, 10-14 September 2006. This paper reviews the theory of Shannon Entropy applied to Monte Carlo source iteration convergence, its implementation, and numerical results.

  3. A Review of Best Practices for Monte Carlo Criticality Calculations, by Forrest Brown, ANS Nuclear Criticality Safety Topical Meeting, Richland, WA, 13-17 September 2009, LA-UR-09-03136. Includes several sample problems: (1) a 2D quarter-core PWR model and (2) a 3D array of steel cans filled with plutonium nitrate. Topics discussed are entropy tests, neutrons per cycle, keff and reaction rate bias, uncertainty bias of keff and reaction rate distributions. Section V lists five specific actions analysts should take.
  4. Status of MORET5 source convergence improvements and benchmark proposal for Monte Carlo depletion calculations, by Y. Richet, W. Haeck, J. Miss, presentation at 2009 WPNCS Source Convergence Expert Group Meeting. Description of source sampling methods in MORET5, including stratified sampling, superhistory, importance sampling, fission matrix, and Wielandt acceleration.

  5. Wielandt Acceleration for MCNP5 Monte Carlo Eigenvalue Calculations, by Forrest Brown, Joint International Topical Meeting on Mathematics & Computation and Supercomputing in Nuclear Applications (M&C + SNA 2007), Monterey, California, 15-19 April 2007. Theory and implementation of Wielandt acceleration, including computational strategy, numerical results, and identification of open questions.

  6. The Sandwich Method for Determining Source Convergence in Monte Carlo Calculations, by J. Yang and Y. Naito, Proc. 7th Intl. Conf. Nuclear Criticality Safety, ICNC2003, Tokai-mura, Iburaki, Japan, 20-24 October 2003, JAEIR-CONF 2003-019, 352 (2003). This is a powerful computational strategy that works without advanced techniques such as Wielandt acceleration or entropy tests (although they make this strategy more powerful).

Additonal information

Source Convergence Bibliography (last updated 2008).

Last updated: 20 March 2013