An infinite-medium analysis is performed for neutron transport spatial discretization methods in planar geometry. Angular flux solutions of the spatially continuous transport equation, which are driven by a linear (or quadratic) source, are shown to vary linearly (or quadratically) in space and angle; these are used to assess whether the discretized transport equations preserve certain cell-averaged and edge quantities. Each of the continuous angular flux solutions has a scalar flux that satisfies the standard diffusion equation; our analysis predicts whether the transport discretizations yield an accurate diffusion coefficient and (diffusion) spatial differencing scheme.The linear moment-based discretization methods under consideration, which are found to preserve certain features of the linear (or quadratic) infinite-medium angular flux solutions, are the familiar linear discontinuous (LD), lumped linear discontinuous (LLD), and linear characteristic (LC) schemes. The step characteristic scheme, which yields an unphysically large diffusion coefficient, is revisited and shown to possess, for diffusive problems, a solution error that would occur if an unphysical anisotropic scattering term had been included in the starting discretized $${S_N}$$SN transport equations.The numerical results verify the theoretical predictions and demonstrate the accuracy of the LD, LLD, and LC schemes in highly scattering problems that are optically thick. Our numerical results also illustrate the impact of inaccuracies in the diffusion coefficient on the numerical solutions of eigenvalue problems. The analysis in this paper has practical implications in the choice of spatial schemes used to solve realistic eigenvalue problems.

The quasidiffusion (QD) method is an established and efficient iterative technique for solving particle transport problems. Each QD iteration consists of a high-order SN sweep, followed by a low-order QD calculation. QD has two defining characteristics: (1) its iterations converge rapidly for any spatial grid and (2) the converged scalar fluxes from the high-order SN sweep and the low-order QD calculation differ, by spatial truncation errors, from each other and from the scalar flux solution of the SN equations. In this paper, we show that by including a transport consistency factor in the low-order equation, the converged high-order and low-order scalar fluxes become equal to each other and to the converged SN scalar flux. However, the inclusion of the transport consistency factor has a negative impact on the convergence rate. We present numerical results that demonstrate the effect of the transport consistency factor on stability.

The coarse-mesh finite difference (CMFD) method is commonly used to accelerate the iterative convergence of single-physics neutron transport problems. For multiphysics problems, the neutron cross sections depend on the temperature and density, both of which depend on the fission heat source; the resulting nonlinear feedback can significantly degrade the performance of CMFD and even cause instability. In this paper, we propose, for a class of one-dimensional (1-D) model multiphysics problems, a new nonlinearly implicit low-order (NILO) CMFD (NILO-CMFD) acceleration method to improve the performance of CMFD-based methods for solving loosely coupled multiphysics problems. Our numerical testing and Fourier analysis show that for the 1-D model problems, the new NILO-CMFD method achieves the same rapid convergence rate that CMFD achieves for single-physics problems.

Abstract In this paper, the standard multigroup neutron diffusion equations are derived as an asymptotic approximation to the multigroup neutron transport equations. The asymptotic analysis employs a scaling that (1) is suggested by the multigroup neutron diffusion equations themselves and (2) generalizes the long-known asymptotic scaling for monoenergetic transport problems. Two other asymptotic scalings of the multigroup transport equations are also considered, both of which lead to a new “group-collapsed” (monoenergetic) “equilibrium” diffusion approximation. The standard multigroup and equilibrium diffusion approximations are shown to preserve certain nonasymptotic properties of the multigroup transport equations. Generalizations of the analyses in this paper, and possible practical applications, are discussed.

fluxes from the Boltzmann transport equation. In Chapter 3, the 2D/1D methodology is described, together with the description of the CMFD iteration process involving dynamic homogenization and solution of the multigroup CMFD linear system. A description of the MPACT depletion algorithm is given in Chapter 4, followed by a discussion of the subgroup and ESSM resonance processing methods in Chapter 5. The final Chapter 6 describes a simplified thermal hydraulics model in MPACT.

The methods presented in this paper solve the Simplified Spherical Harmonics approximation to the multidimensional neutron transport equation. 1D, 2D and 3D systems were modeled with Cartesian geometry using the finite difference method to discretize the spatial variables. The method is able to simulate any energy group discretization, including up-scattering terms. The Krylov Shur method was used to calculate the solution of the steady-state equation by solving a generalized eigenvalue problem. This methodology has the capability to calculate any number of eigenfunctions. A formulation review of the Simplified Spherical Harmonics is explained in this work, as well as, a study of the boundary conditions for different approaches of the finite difference method. The results calculated by this methodology are compared with the discrete ordinates and diffusion approximation methods, all of them, using the same spatial discretization in order to show the different accuracy of each method without influence of the method used for discretizing the spatial variable. The results show the validity of each method for different benchmark problems.

Coarse Mesh Finite Difference (CMFD) is a widely-used iterative acceleration method for neutron transport problems in which nonlinear terms are introduced in the derivation of the low-order CMFD diffusion equation. These terms, including the homogenized diffusion coefficient, the current coupling coefficients, and the multiplicative prolongation constant, are subject to numerical instability when a scalar flux estimate becomes sufficiently small or negative. In this paper, we use a suite of contrived problems to demonstrate the susceptibility of CMFD to failure for each of the vulnerable quantities of interest. Our results show that if a scalar flux estimate becomes negative in any portion of phase space, for any iterate, numerical instability can occur. Specifically, the number of outer iterations required for convergence of the CMFD-accelerated transport problem can increase dramatically, or worse, the iteration scheme can diverge. An alternative Linear Diffusion Acceleration (LDA) scheme addresses these issues by explicitly avoiding local nonlinearities. Our numerical results show that the rapid convergence of LDA is unaffected by the very small or negative scalar flux estimates that can adversely affect the performance of CMFD. Therefore, our results demonstrate that LDA is a robust alternative to CMFD for certain sensitive problems in which CMFD can exhibit reduced effectiveness or failure.

A second-order “Time-Dependent Multiple Balance” (TDMB) method for solving neutron transport problems is introduced and investigated. TDMB consists of solving two coupled equations: (i) the original balance equation (the transport equation integrated over a time step) and (ii) the “balance-like” auxiliary equation (an approximate neutron balance equation). Simple analysis shows that TDMB is second-order accurate and robust (unconditionally free from spurious oscillation). A source iteration (SI) method with diffusion synthetic acceleration (DSA) is formulated to solve these equations. A Fourier analysis reveals that the convergence rates of the proposed iteration schemes for TDMB are similar to those of the common (SI + DSA) schemes for Backward Euler (BE); however, TDMB requires about twice the computational effort per iteration. To demonstrate the theory—accuracy, robustness, and convergence rate—and investigate the efficiency of TDMB, we present results from a discrete ordinates (Sn) research code. Results are discussed, and future work is proposed.

Two-dimensional/one-dimensional (2D/1D) methods have become popular for solving the 3D Boltzmann neutron transport equation on medium-to-large computing platforms. These methods can have a wide range of accuracy that depends largely on the fidelity of the coupling between the 2D and 1D solutions in the spatial and angular variables. In general, methods with higher-order coupling are both more accurate and more computationally expensive. In order to simplify and reduce computation, an isotropic angular coupling term is frequently used. The deficiency of this approximation compared to higher-order angular coupling has been studied experimentally, but there is insufficient theoretical analysis in the literature to supplement the experimental results. In this paper, an asymptotic analysis is applied to the 2D/1D equations with varying orders of angular coupling to facilitate comparison to the simplified PN (SPN) equations. We find that the 2D/1D method with 3 angular coupling moments preserves the 3D SP3 limit, while the 2D/1D method with isotropic coupling does not. As a result, the isotropic coupling method is theoretically less accurate in problems with strong spatial gradients in both the radial and axial dimensions. This analysis provides a theoretical basis for design and optimization of the angular coupling scheme in a 2D/1D method. The results of the theoretical analysis are confirmed by using the Takeda-Ikeda benchmark to compare the accuracy of 2D/1D methods with isotropic and anisotropic coupling implemented in MPACT to SP1 and SP3 finite difference solutions.

A new method for calculating anisotropic radial transverse leakage (TL) in a two-dimensional (2D)/one-dimensional (1D) transport method is derived and implemented in MPACT. This method makes use of parity in the polar angle only to form the 2D transport equations for the 2D/1D method. The even-parity component is solved on a fine mesh using the method of characteristics (MOC), while the odd-parity component is solved on a coarse mesh using S. The anisotropic radial TL on the coarse cell boundaries is calculated by combining the even- and odd-parity components. The new method is faster than a similar previous method because it delegates half of the work required to calculate the solution of the 2D transport problem to a coarse-mesh S solver, which is more than ten times faster than the fine-mesh MOC solver. The results show that the accuracy of the new method is equivalent to that of the previously implemented method for anisotropic TL, with a significant speedup. With azimuthally isotropic TL, the new method reduces the computational overhead compared to the standard method from 58% to 5% for the three-dimensional (3D) C5G7 benchmark problems. With azimuthally anisotrop\ic TL using Fourier expansion, the new method reduces the overhead from 84% to 37%. This is important because the accuracy of the 2D/1D method is limited by the isotropic TL approximation. With anisotropic TL, the accuracy of 2D/1D is equivalent or comparable to 3D transport, but there is a significant computational cost associated with calculating the anisotropic TL. The method presented provides a faster way to calculate the anisotropic TL, giving the 2D/1D method significantly increased accuracy with only a modest increase in computational requirements compared to isotropic 2D/1D.

The Multilevel in Space and Energy Diffusion (MSED) method accelerates the iterative convergence of multigroup diffusion eigenvalue problems by performing work on lower-order equations with only one group and/or coarser spatial grids. It consists of two primary components: (1) a grey (one-group) diffusion eigenvalue problem that is solved via Wielandt-shifted power iteration (PI) and (2) a multigrid-in-space linear solver. In previous work, the efficiency of MSED was verified using Fourier analysis and numerical results from a one-dimensional multigroup diffusion code. Since that work, MSED has been implemented as a solver for the coarse-mesh finite difference (CMFD) system in the three-dimensional Michigan Parallel Characteristics Transport (MPACT) code. In this paper, the results from the implementation of MSED in MPACT are presented, and the changes needed to make MSED more suitable for MPACT are described. For problems without feedback, the results in this paper show that MSED can reduce the CMFD run time by an order of magnitude and the overall run time by a factor of 2 to 3 compared to the default CMFD solver in MPACT [PI with the generalized minimal residual (GMRES) method]. For problems with feedback, the convergence of the outer Picard iteration scheme is worsened by the well-converged CMFD solutions produced by the standard MSED method. To overcome this unintuitive deficiency, MSED may be run with looser convergence criteria (a modified version of the MSED method called MSED-L) to circumvent the issue until the multiphysics iteration in MPACT is improved. Results show that MSED-L can reduce the CMFD run time in MPACT by an order of magnitude, without negatively impacting the outer Picard iteration scheme.

The Two-Dimensional (2-D)/One-Dimensional (1-D) method allows pin-resolved computational transport solutions for large, full-core light water reactor simulations at relatively low computational cost compared to a true three-dimensional (3-D) transport method. The 2-D/1-D method constructs an approximation to the 3-D transport equation with (1) a 2-D transport equation in the radial variables x and y, discretized on a fine radial spatial grid, and (2) a 1-D transport (or approximate P-N) equation in the axial variable z, discretized on a radially coarse spatial grid. The 2-D and 1-D equations are coupled through transverse leakage (TL) terms. In this paper, a new 2-D/1-D P-3 method with anisotropic transverse leakages and anisotropic homogenized 1-D cross sections (XSs) is proposed to improve the accuracy of conventional 2-D/1-D with pin homogenization. It is shown that only the polar component of the anisotropic homogenized XS has a significant effect on the solution; the azimuthal component is negligible. However, the polar and azimuthal components of the leakage terms are both important. The new method is implemented in the 2-D/1-D code Michigan PArallel Characteristics Transport (MPACT). The method in this paper is shown to achieve nearly 3-D transport accuracy with sufficient refinement in space and angle. The improvement of this new method compared to the previous 2-D/1-D method in MPACT is most notable in problems with strong axial leakage and sharp axial discontinuities, such as control rod tips or part-length rods. The method is computationally more expensive than the existing 2-D/1-D method with isotropic TL and XSs, but this additional cost may be justified when the axial flux shape does not vary smoothly due to axial heterogeneity and needs to be resolved well.

In this paper, we present a new multilevel in space and energy diffusion (MSED) method for solving multigroup diffusion eigenvalue problems. The MSED method can be described as a PI scheme with three additional features: (1) a grey (one-group) diffusion equation used to efficiently converge the fission source and eigenvalue, (2) a space-dependent Wielandt shift technique used to reduce the number of PIs required, and (3) a multigrid-in-space linear solver for the linear solves required by each PI step. In MSED, the convergence of the solution of the multigroup diffusion eigenvalue problem is accelerated by performing work on lower-order equations with only one group and/or coarser spatial grids. Results from several Fourier analyses and a one-dimensional test code are provided to verify the efficiency of the MSED method and to justify the incorporation of the grey diffusion equation and the multigrid linear solver. These results highlight the potential efficiency of the MSED method as a solver for multidimensional multigroup diffusion eigenvalue problems, and they serve as a proof of principle for future work. Our ultimate goal is to implement the MSED method as an efficient solver for the two-dimensional/three-dimensional coarse mesh finite difference diffusion system in the Michigan parallel characteristics transport code. The work in this paper represents a necessary step towards that goal.

The lower bounds for the theoretical convergence rate of variants of the Coarse Mesh Finite Difference (CMFD) method for neutron transport acceleration are studied in this paper by generalization of the method into three categories: artificially diffusive CMFD, flux relaxation, and higher-order spatial prolongation operators. A Fourier analysis of the methods demonstrates that artificial diffusion and flux relaxation are mathematically equivalent and arbitrarily scale the coarse mesh to fine mesh projection (CMP) vector. The high-order spatial prolongation method is shown to affect the shape of the CMP vector. As a result, any of the CMFD variants based on these three sets of modifications correspond to a specific CMP vector. The optimization process is performed for the multidimensional vector, and the minimum spectral radius among all possible CMP vectors is shown to be the theoretical lower bound for the CMFD convergence rate. The spectral radius associated with the CMFD convergence rate lower bound is found to be slightly smaller (less than 0.04) than optimally diffusive CMFD(odCMFD), and the difference between odCMFD to the CMFD lower bound is much smaller than the difference between both standard CMFD and partial current-based CMFD to the CMFD lower bound. In addition, the odCMFD method has a distinct advantage in ease of implementation and minimal overhead. Conversely, the implementation necessary to achieve the CMFD lower bound would be very complicated, especially for two-and three-dimensional problems.

In this work, we derive and test variational discontinuity factors (DFs) for the asymptotic homogenized diffusion equation. We begin with a functional for optimally estimating the reactor multiplication factor, then introduce asymptotic expressions for the forward and adjoint angular fluxes, and finally require that all first-order error terms vanish. In this way, the reactor multiplication factor can be calculated with second-order error. The analysis leads to (1) an alternate derivation of the asymptotic homogenized diffusion equation, (2) variational boundary conditions for large periodic systems, and (3) variational DFs to be applied between adjacent periodic regions (e.g., fuel assemblies). Numerical tests show that applying the variational DFs to the asymptotic homogenized diffusion equation yields the most accurate estimates of the reactor multiplication factor compared to other DFs for a wide range of problems. However, the resulting assembly powers are less accurate than those obtained using other DFs for many realistic problems.

Abstract This paper presents a systematic derivation of space-dependent diffusion coefficients for neutronically diffusive 3D systems with anisotropic scattering that contain “small" voided subregions. The resulting diffusion coefficients in a solid homogeneous part of the physical system reduce to the standard D = 1/3Σt. However, for spatial points in or near a void region, the diffusion coefficient becomes a space-dependent 3×3 anisotropic tensor, which can be obtained by solving a 3D transport problem without scattering. The resulting anisotropic diffusion theory should be valid if the void regions are sufficiently small that they perturb, but do not significantly alter, the “diffusive" character of the system.

The 2D/1D method is an important tool in LWR analysis that enables pin-resolved transport solutions for large, full-core simulations at relatively low computational cost compared to a true 3D transport method. To use the 2D/1D approximation, several assumptions are made about the nature of the axial (z) component of the true 3D transport solution. Generally, it is assumed that the axial variation of the solution is weak enough that the axial streaming term can be approximated as isotropic in angle, and uniform in space over a coarse pin cell, without unacceptable detriment to accuracy. Also, it is assumed that a homogenized 1D axial pin cell calculation will give the correct axial streaming magnitude and power shape. In some cases, however, we desire a more accurate transport solution; we can achieve this by implementing higher-fidelity solvers that avoid some of these coarse approximations, e.g. allowing angular dependence in the radial and axial leakage terms that couple the 2D and 1D solutions. Since the introduction of 2D/1D methods, there has been interest in developing and improving these more accurate approximations, with the ultimate goal being a 2D/1D method that limits to the 3D transport solution with spatial and angular refinement. In this paper, the 2D/1D code MPACT is used, with SN as the 1D axial solver. We evaluate some of the approximations made in MPACT, and derive a new (to MPACT) angle-dependent 2D to 1D total cross section homogenization. The new homogenization shows good results for C5G7-type problems and the full 3D C5G7 benchmarks compared to what was previously the “most accurate” 2D/1D method in MPACT, which used angle-dependent leakages but only scalar flux homogenization of the total cross section. We also quantify the effects of applying a within-pin fine-mesh spatial shape to the axial transverse leakage, which is generally much smaller than the effect of angle-dependent homogenization.