Hybrid Two-Level Transport Method with Solution Decomposition in Macro and Micro Components

Technical Summary: Hybrid Two-Level Transport Method with Solution Decomposition in Macro and Micro Components

Problem Statement
The paper addresses the computational challenge of solving the one-group steady-state Boltzmann transport equation. While deterministic methods offer convergence with mesh refinement but suffer from discretization errors, stochastic Monte Carlo (MC) methods introduce only statistical noise that decreases with particle count but can be computationally expensive due to variance. The authors aim to develop a hybrid MC/deterministic approach that improves computational efficiency and reduces variance by decomposing the transport solution into macroscopic and microscopic components.

Methodology
The proposed method, termed the Hybrid Micro-Macro Transport (HMMT) method, relies on decomposing the angular flux $\psi(x, \mu)$ into a macro component $\Psi(x, \mu)$ and a micro component $\delta\psi(x, \mu)$:
$$\psi(x, \mu) = \Psi(x, \mu) + \delta\psi(x, \mu)$$

• Macro Component: This component captures the large-scale structure of the solution. It is defined using the $P_1$ approximation, where the angular flux is reconstructed from the first two angular moments (scalar flux $\phi$ and current $J$). These moments are obtained by solving low-order quasidiffusion (LOQD) equations, also known as Variable Eddington Factor (VEF) equations. The LOQD system utilizes exact closures defined by Eddington factors and boundary factors.
• Micro Component: This component represents the fine-scale deviations from the macro solution. It is governed by a transport equation with a source term derived from the macro component. Crucially, the micro equation contains no scattering term; it is solved using a Monte Carlo simulation.
• Iterative Solution Scheme: The system is solved via a fixed-point iteration scheme (Algorithm 1):
  1. LOQD Solve: Using closure factors ($E, C_L, C_R$) from the previous iteration, the deterministic LOQD equations are solved to update the macro moments ($\phi, J$) and reconstruct $\Psi$.
  2. MC Solve: The micro transport equation is solved using MC. The source term for this step is constructed from the updated macro solution. The MC simulation uses particles with both positive and negative weights to handle the potentially negative source terms arising from the decomposition.
  3. Closure Update: The MC simulation tallies the angular moments of the micro component ($\delta\phi, \delta J$). These are combined with the macro moments to compute updated Eddington and boundary factors for the next iteration.
  4. Convergence: The process repeats until the scalar flux converges within a specified tolerance.

Key Contributions

• Novel Decomposition: The paper introduces a specific hybrid framework where the macro component is strictly defined by $P_1$-approximated moments derived from LOQD equations, while the micro component is treated as a non-scattering MC problem.
• Exact Closures: The method employs exact closures for the low-order equations, where the Eddington factors are dynamically calculated based on the sum of the macro and micro components.
• Variance Reduction Mechanism: By solving the dominant large-scale behavior deterministically and using MC only for the residual micro component, the method aims to reduce the statistical noise inherent in pure MC simulations.

Numerical Results
The method was tested on a heterogeneous slab problem featuring an optically thick barrier, comparing the HMMT results against a traditional MC solution with implicit capture (IC).

• Accuracy: The HMMT solution demonstrated higher accuracy than the traditional MC solution, particularly in the source region and the subdomain behind the optically thick barrier. The $L_2$-norm of the relative error was 0.48 for HMMT compared to 1.3 for MC.
• Variance Reduction: The batch relative variance was significantly reduced in the source region and the region behind the barrier. The HMMT solution exhibited approximately two orders of magnitude less variance in the 2-norm compared to the standard MC approach.
• Convergence: The method achieved near-theoretical convergence rates ($O(N^{-1/2})$) with respect to the number of particles ($N$) and $O(\Delta x)$ with respect to mesh size ($\Delta x$). At high particle counts, the error plateaued due to discretization error, consistent with theoretical expectations.
• Efficiency (Figure of Merit): The Figure of Merit (FOM) for the HMMT method was found to be over 100 times greater than that of the traditional MC method. This improvement is attributed to both reduced variance and reduced runtime, as the MC portion of the HMMT method involves no scattering events, which are typically the most computationally expensive part of transport simulations.

Significance and Claims
The authors claim that the HMMT method successfully demonstrates a reduction in the stochastic numerical solution's variance and a significant improvement in computational efficiency compared to standard MC algorithms with implicit capture. The paper positions this work as a new hybrid approach for steady-state transport problems. The authors note that future research will involve analyzing the method on various test problems, exploring different definitions of the macro component, applying other types of low-order equations, and investigating advanced discretization schemes. The work was supported by the Center for Advancing the Radiation Resilience of Electronics (CARRE).