Hybrid Weight Window Method for Global Time-Dependent Monte Carlo Particle Transport Calculations
Abstract
This paper presents a new Monte Carlo (MC) algorithm for time-dependent particle transport problems with global variance reduction based on automatic weight windows (WWs). The centers of WWs at a time step are defined by the solution of an auxiliary hybrid MC / deterministic problem formed by the low-order second-moment (LOSM) equations. The closures for the hybrid LOSM equations are calculated by the MC method. The LOSM equations are discretized by a scheme of the second-order accuracy in time and space. Filtering techniques are applied to reduce noise effects in the LOSM closures. The WWs defined with the auxiliary solution give rise to sufficiently uniform MC particle distribution in space on each time step. The algorithm is analyzed by means of an analytic transport benchmark. We study performance of the MC algorithm depending on a set parameters of WWs. Figure of merit and relative error results are presented, demonstrating the performance of the hybrid MC method and quantifying its computational efficiency.
Summary
This paper introduces a new hybrid Monte Carlo (MC) algorithm designed for time-dependent particle transport problems. The core innovation is a global variance reduction technique that leverages automatic weight windows (WWs). These WWs are dynamically adjusted at each time step, with their centers determined by the solution of an auxiliary hybrid MC/deterministic problem. This auxiliary problem is formulated using low-order second-moment (LOSM) equations, where the necessary closures for these equations are estimated using the MC method itself. The LOSM equations are discretized with a second-order accurate scheme, and filtering techniques are applied to mitigate noise in the LOSM closures. The resulting WWs aim to create a more uniform distribution of MC particles in space at each time step, thereby improving the efficiency of the MC simulation. The paper validates the algorithm's performance using an analytic transport benchmark, analyzing its sensitivity to various WW parameters and presenting figure of merit (FOM) and relative error results to demonstrate its computational efficiency. This matters to the field as it offers a novel approach to variance reduction in time-dependent particle transport, a critical challenge in many nuclear engineering and physics applications. The authors develop a hybrid approach combining the strengths of Monte Carlo and deterministic methods. By using the LOSM equations to guide the weight windows, the algorithm can achieve better global variance reduction than traditional methods that rely on adjoint solutions. The use of filtering techniques to reduce noise in the LOSM closures is also a key contribution, as it allows for more accurate and stable weight window updates. The detailed analysis of the algorithm's performance and sensitivity to various parameters provides valuable guidance for practitioners looking to implement this method in their own simulations. Ultimately, this research advances the state-of-the-art in Monte Carlo particle transport by offering a more efficient and robust approach for solving time-dependent problems with global variance reduction.
Key Insights
- •A hybrid MC/deterministic approach using LOSM equations provides a more efficient method for global variance reduction in time-dependent particle transport compared to traditional adjoint-based methods.
- •The method uses a second-order accurate scheme for discretizing the LOSM equations in time and space.
- •Filtering techniques (moving average and Fourier filtering) applied to the LOSM closures and initial conditions significantly reduce noise effects and improve the accuracy of the weight windows.
- •The algorithm's performance is highly sensitive to the choice of weight window parameters, particularly `ε_min` (minimum weight window center) and `ρ` (window width). Decreasing `ε_min` improves wave front resolution but can lead to excessive particle splitting.
- •Updating the weight windows too frequently during a time step can lead to instabilities in the Monte Carlo solution. An optimal balance between update frequency and statistical error in the closures is crucial.
- •The hybrid solution computed with unfiltered quantities exhibited relative L2 error smaller than the Monte Carlo solution from the same particle histories. Application of filtering techniques further reduced this error.
- •Compared to Lagged Weight Windows (LWW), the Hybrid Weight Window (HWW) algorithm provides a more uniform relative variance and enables efficient solutions in low-flux regions.
Practical Implications
- •The algorithm can be applied to a wide range of time-dependent particle transport problems, including reactor physics, radiation shielding, and medical physics.
- •Researchers and engineers can use this algorithm to improve the efficiency and accuracy of their Monte Carlo simulations, particularly for problems where global variance reduction is critical.
- •Practitioners can leverage the parameter sensitivity analysis provided in the paper to optimize the algorithm's performance for specific applications. The optimal choice of parameters, such as ε_min and ρ, depends on the specific problem being solved.
- •The algorithm can be integrated with multiphysics codes, where material properties and sources are evaluated at each Monte Carlo census, allowing for more accurate and realistic simulations.
- •Future research directions include extending the algorithm to multi-dimensional problems, developing more efficient tallying methods for the hybrid problem, and exploring monotonization techniques for discretizing the LOSM equations.