Diffusion Synthetic Acceleration Research Articles

A Variant of the Second-Moment Method for k-Eigenvalue Calculations

The second-moment (SM) method is a linear variant of the quasi-diffusion (QD) method for accelerating the iterative convergence of Sn source calculations. It has several significant advantages relative to the QD method, diffusion synthetic acceleration, and nonlinear diffusion acceleration. Here, we define a variant of this method for k-eigenvalue calculations that retains the advantages of the original method, and we computationally demonstrate the efficacy of the method for simple example calculations. In particular, this method has two important properties. First, it is a linear acceleration scheme requiring only the solution of a pure k-eigenvalue diffusion equation with a corrective source term as opposed to a k-eigenvalue drift-diffusion equation. Second, unconditional stability is achieved even when the diffusion equation is not discretized in a manner consistent with the Sn spatial discretization. We are unaware of any other scheme that has these properties. We also show a connection between our method and the k-eigenvalue acceleration technique of Barbu and Adams, which motivated us to develop our SM method.

Revisit interior penalty based diffusion synthetic acceleration for the SN transport equation discretized with discontinuous Galerkin method

Diffusion Synthetic Acceleration (DSA) techniques based on symmetric and nonsymmetric versions of the Interior Penalty (IP) method have been proposed to accelerate the scattering source iteration of the SN transport equation discretized with discontinuous Galerkin (DG) method in the previous researches. The performances of DSA based on various IP methods need to be compared comprehensively. The Robin boundary condition is usually adopted to treat the vacuum boundaries in solving the diffusion equation of DSA, but in the thick diffusive limit, the Dirichlet boundary condition was derived for the diffusion equation to give the correct leading-order interior solution. The influence of the boundary condition for the vacuum boundaries in DSA needs to be analyzed. In this study, the DSA schemes with three various IP methods (SIP, IIP and NIP) in conjunction with Krylov iterative method are summarized in a complete view, and the approaches to dealing with the instability problem of DSA in the optically thick and nearly voided regimes are discussed. From the numerical experiments, it suggests that the DSA schemes with IIP, NIP and SIP perform closely to each other, and there is no specific IP method which shows advantages over the others under any conditions. With other conditions fixed, the performance of the DSA scheme with the Dirichlet boundary condition becomes better than that with the Robin boundary condition with the increase of optical cell thickness. There is no apparent relationship between the performance of the DSA scheme and the discretization order of DG.

A diffusion synthetic acceleration method for steady discrete unified gas kinetic scheme in radiative heat transfer

An efficient synthetic-type iterative method is proposed to improve the convergence rate of the steady discrete unified gas kinetic scheme (SDUGKS) for steady radiative heat transfer problems. The key ingredient of the present method is the tight coupling of the mesoscopic radiative transfer equation (RTE) and the macroscopic diffusion equation. The RTE is calculated by SDUGKS, and the diffusion equation is solved to predict the incident radiation energy and angular radiation intensity used in SDUGKS to accelerate the evolution, unlike the original diffusion synthetic acceleration (DSA) method which only predicts the incident radiation energy. Furthermore, a damping factor is adopted to remedy the inconsistency of spatial discretization between the mesoscopic SDUGKS and macroscopic diffusion equation. The theoretical convergence rate of the present method is studied via the discrete Fourier analysis. Several numerical tests under a wide range of optical thicknesses are performed to show the convergence speed. The damping factor and the correction of the angular radiation intensity added in solving the macroscopic diffusion equation greatly increase the stability of the present method and also reduce its computational costs. The convergence rate of the present method becomes faster than that of previous SDUGKS when the optical thickness is larger than 10. With the increase of extinction coefficient, the speedup ratio of the present method and SDUGKS in one-dimensional problems becomes extremely large and also reaches one or two orders of magnitude in two- and three-dimensional problems.

A hybrid neutronics method with novel fission diffusion synthetic acceleration for criticality calculations

A novel Fission Diffusion Synthetic Acceleration (FDSA) method is developed and implemented as a part of a hybrid neutronics method for source convergence acceleration and variance reduction in Monte Carlo (MC) criticality calculations. The acceleration of the MC calculation stems from constructing a synthetic operator and solving a low-order problem using information obtained from previous MC calculations. By applying the P1 approximation, two correction terms, one for the scalar flux and the other for the current, can be solved in the low-order problem and applied to the transport solution. A variety of one-dimensional (1-D) and two-dimensional (2-D) numerical tests are constructed to demonstrate the performance of FDSA in comparison with the standalone MC method and the coupled MC and Coarse Mesh Finite Difference (MC-CMFD) method on both intended purposes. The comparison results show that the acceleration by a factor of 3–10 can be expected for source convergence and the reduction in MC variance is comparable to CMFD in both slab and full core geometries, although the effectiveness of such hybrid methods is limited to systems with small dominance ratios.

Convergence Properties of a Linear Diffusion-Acceleration Method for k-Eigenvalue Transport Problems

Most methods that use low-order operators to accelerate the iterative solution of transport eigenvalue problems employ nonlinear functionals of the transport solution (such as Eddington tensors) in their low-order equations, which are themselves standard eigenvalue problems. Here, we discuss linear diffusion synthetic acceleration (DSA) for -eigenvalue problems, which belongs to a family of methods that has received less attention than its nonlinear counterparts. We review the history of these linear methods as far as we know it and describe theoretical questions that to our knowledge have remained unanswered. With these methods, a low-order problem is solved after each transport step for an updated eigenvalue and an additive correction to the eigenfunction. These low-order problems are not standard eigenvalue problems, for they contain residuals as fixed sources. The low-order problems admit infinitely many solutions (updated and additive correction to the eigenfunction), and the solution that is obtained depends on the initial guess and iterative method chosen for the low-order problems. Experience has shown that when the low-order problems are solved with a powerlike iteration method and certain initial guesses, they yield solutions that cause rapid convergence to the correct high-order solution. We study the convergence properties of this algorithm applied to two model problems: an infinite homogeneous medium and a one-cell problem. For the infinite homogeneous problem, we present a Fourier analysis of the linear DSA method, which demonstrates that when the low-order problems are solved using a powerlike iteration scheme, the linear DSA scheme provides immediate convergence of the -eigenvalue and rapid convergence of the eigenfunction (much like DSA applied to scattering iterations in fixed-source problems). For the one-cell problems, we find that the linear scheme for -eigenvalue problems performs approximately as well as DSA for fixed-source problems. The latter analysis reveals a quantitative bound on the consistency between low- and high-order operators that is necessary and sufficient for convergence of those problems. With some theoretical foundations for the linear methods now established, we turn to numerical testing. We find, as others have before us using different low-order operators, that the method works well in practice. We provide numerical results from reactor problems in which our linear DSA is approximately as effective as the more widely used nonlinear methods. Our theoretical and numerical results add to the body of evidence that the linear methodology offers a simple path to rapid convergence of -eigenvalue problems, especially for codes that already employ linear low-order operators to converge scattering iterations.

Accelerated Deterministic Phonon Transport With Consistent Material Temperature and Intensities

Abstract We present a method for deterministically solving the frequency and temperature dependent phonon radiative transport (PRT) equation in the single-mode relaxation time (SMRT) approximation in the self-adjoint angular flux (SAAF) form. To handle the nonlinear coupling between the phonon intensities and the material temperature, we apply a linearization approach that is similar to one in thermal radiative transport. This procedure leads to the PRT equation with pseudo-scattering. The method presented includes acceleration of both the inner pseudo-scattering source iterations and outer temperature iteration with a gray diffusion synthetic acceleration (DSA) and Anderson acceleration, respectively. We use the finite-element method to discretize the PRT equation in space and the method of discrete ordinates (SN) for angular discretization. The proposed method is verified by a gray method of manufactured solutions problem and demonstrated on a problem using temperature and direction dependent multigroup data from lithium aluminate (LiAlO2). The iterative performance of the acceleration method in each test is then compared to the unaccelerated method.

Anderson acceleration and linear diffusion for accelerating the [formula omitted]-eigenvalue problem for the transport equation

This work focuses on the k-eigenvalue problem of the neutron transport equation to solve for the largest eigenvalue (keff) and the corresponding eigenmode, called the fundamental mode. This problem is usually solved using the power iteration method. However, the convergence of this algorithm can be very slow, especially if the dominance ratio is high as is the case in reactor physics applications. Thus, the power iteration method has to be accelerated in some ways to improve its convergence.One such acceleration is the Chebyshev acceleration method which has been widely applied to legacy codes. In recent years, nonlinear methods have been applied to solve the k-eigenvalue problem. Nevertheless, they are often compared to the unaccelerated power iteration. Past works have compared and contrasted the benefits of the Anderson acceleration method for the neutron transport equation, and the analysis is improved in this work by studying the impact of preconditioning acceleration techniques on the overall performance.In this work, the transport operator is inverted using the source iteration method with the discrete ordinates (Sn) method. Therefore, keeping in mind the need for improving the power iteration algorithm, we also focus on a different type of acceleration which is the diffusion synthetic acceleration (DSA) usually employed for improving the inner iterations. In this research effort, DSA is applied to the power iteration. Besides, Anderson and Chebyshev accelerations are combined with this external DSA scheme to evaluate the computational gain.All these methods have been applied to a benchmark case of a small pressurised water reactor, KAIST 3A.

STRAUM-MATXST: A code system for multi-group neutron-gamma coupled transport calculation with unstructured tetrahedral meshes

In this paper, a new multi-group neutron-gamma transport calculation code system STRAUM-MATXST for complicated geometrical problems is introduced and its development status including numerical tests is presented. In this code system, the MATXST (MATXS-based Cross Section Processor for SN Transport) code generates multi-group neutron and gamma cross sections by processing MATXS format libraries generated using NJOY and the STRAUM (SN Transport for Radiation Analysis with Unstructured Meshes) code performs multi-group neutron-gamma coupled transport calculation using tetrahedral meshes. In particular, this work presents the recent implementation and its test results of the Krylov subspace methods (i.e., Bi-CGSTAB and GMRES(m)) with preconditioners using DSA (Diffusion Synthetic Acceleration) and TSA (Transport Synthetic Acceleration). In addition, the Krylov subspace methods for accelerating the energy-group coupling iteration through thermal up-scatterings are implemented with new multi-group block DSA and TSA preconditioners in STRAUM.

Revisit Interior Penalty Based Diffusion Synthetic Acceleration for the Sn Transport Equation Discretized with Discontinuous Galerkin Method

Revisit Interior Penalty Based Diffusion Synthetic Acceleration for the Sn Transport Equation Discretized with Discontinuous Galerkin Method

Modified accelerate iteration for optical property reconstruction based on time-domain radiative transfer equation

A modified diffusion synthetic acceleration method is proposed to speed up the convergence process of TD-RTE calculation by employing the finite element method, combined with OpenMP parallel technology to further enhance the computational efficiency. In addition, the sequential quadratic programming algorithm is applied with Gaussian Markov random field (GGMRF) regularization to reconstruct the absorption and scattering parameter fields simultaneously. The accuracy and efficiency of the proposed model are investigated. It was observed that the modified DSA model is more efficient with the increase of the optical thickness, where the acceleration ratio is improved to 6 for the optical thicker media. In addition, similar acceleration effect was witnessed for both scattering dominated and non-scattering dominated media. Furthermore, by employing the modified DSA model into the reconstruction, it is interesting to remark that not only the reconstruction time is significantly reduced with the acceleration ratio of 8–11, but also the reconstructed image is further improved. All the results indicate that the developed scheme is efficient and robust based on TD-RTE, showing great potential in optical imaging.

Hybrid approaches for accelerated convergence of block-Jacobi iterative methods for solution of the neutron transport equation

The Parallel Block Jacobi (PBJ) [1,2] scheme is an iterative method with application in neutron transport solution which requires asynchronous sub-domain interfaces, where asynchronous here means flux continuity is not imposed across sub-domain interfaces. In this work, we present the development, analysis, and testing of hybrid methods which utilize both PBJ-type methods and synchronous sweep-based methods in a single iterative step. These methods are developed to reduce the penalty to iterative performance of the former incurred in problems containing optically thin cells, while maintaining the PBJ method's applicability to massively parallel solution on unstructured grids. As a prelude to this ultimate goal, this study is conducted in serial operation on 2-D Cartesian grids to facilitate implementation of the various tested iterative methods, to enable the reported spectral analysis, and for shorter computational execution times allowing the large amounts of performance data to be collected. The analysis consists of two Fourier analyses, demonstrating the sequential execution of a PBJ method and SI to be iteratively robust with respect to optical thickness. The results of these analyses and preliminary computational experimentation demonstrate the effectiveness of methods that execute PBJ and SI only in optically thick and thin regions of a problem, respectively. Further experimentation, including a parametric study, then demonstrates this approach to be effective when the thin region is solved using Inexact Parallel Block Jacobi (IPBJ), with sub-domains of a size such that the applicability to massively parallel solution on unstructured grids is not diminished. Finally, testing on realistic test problems containing large void regions confirms the viability of this approach. Performance comparisons include the traditional iterative acceleration methods Diffusion Synthetic Acceleration, Partial-current Nonlinear Diffusion Acceleration, and Adjacent-cell Preconditioner. The results show that while the hybrid PBJ methods generally consume more iterations than these traditional methods, our hybrid approach maintains an iteration count on the same order of magnitude as these acceleration methods, which are not fully developed for massively parallel solution on unstructured grids.

Modified Fokker-Planck Acceleration for Forward-Peaked Transport Problems in Slab Geometry

This paper introduces a new acceleration technique for the convergence of the solution of transport problems with highly forward-peaked scattering. The technique is similar to a conventional high-order/low-order (HOLO) acceleration scheme. The Fokker-Planck equation, which is an asymptotic limit of the transport equation in highly forward-peaked settings, is modified and used for acceleration; this modified equation preserves the angular flux and moments of the (high-order) transport equation. We present numerical results using the Screened Rutherford, Exponential, and Henyey–Greenstein scattering kernels and compare them to established acceleration methods such as diffusion synthetic acceleration (DSA). We observe three to four orders of magnitude speed-up in wall-clock time compared to DSA.

A ROBUST SECOND-ORDER MULTIPLE BALANCE METHOD FOR TIME-DEPENDENT NEUTRON TRANSPORT SIMULATIONS

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.

ACCELERATION OF A 2-DIMENSIONAL, 2-ENERGY GROUP NEUTRON NOISE SOLVER BASED ON A DISCRETE ORDINATES METHOD IN THE FREQUENCY DOMAIN

The acceleration of the convergence rate is studied for a neutron transport solver to simulate 2-D, 2-energy-group neutron noise problems in the frequency domain. The Coarse Mesh Finite Difference (CMFD) method is compared to the Diffusion Synthetic Acceleration (DSA) method. Numerical tests are performed using heterogeneous system configurations with different boundary conditions. The CMFD scheme leads to a better convergence rate. The results also show that CMFD can accelerate neutron noise problems in a wide range of perturbation frequencies with almost equal efficiency. An unstable convergence behavior is nevertheless observed in problems with purely reflective boundary conditions. Stabilization techniques such as performing multiple transport sweeps, underrelaxing the flux update, and using the lpCMFD method are investigated and improvements can be obtained.

Diffusion Synthetic Acceleration for Heterogeneous Domains, Compatible with Voids

A standard approach to solving the S transport equations is to use source iteration with diffusion synthetic acceleration (DSA). Although this approach is widely used and effective on many problems, there remain some practical issues with DSA preconditioning, particularly on highly heterogeneous domains. For large-scale parallel simulation, it is critical that both (a) preconditioned source iteration converges rapidly and (b) the action of the DSA preconditioner can be applied using fast, scalable solvers, such as algebraic multigrid (AMG). For heterogeneous domains, these two interests can be at odds. In particular, there exist DSA diffusion discretizations that can be solved rapidly using AMG, but they do not always yield robust/fast convergence of the larger source iteration. Conversely, there exist robust DSA discretizations where source iteration converges rapidly on difficult heterogeneous problems, but fast parallel solvers like AMG tend to struggle applying the action of such operators. Moreover, very few current methods for the solution of deterministic transport are compatible with voids. This paper develops a new heterogeneous DSA preconditioner based on only preconditioning the optically thick subdomains. The resulting method proves robust on a variety of heterogeneous transport problems, including a linearized hohlraum mesh related to inertial confinement fusion. Moreover, the action of the preconditioner is easily computed using AMG iterations, convergence of the transport iteration typically requires 2 to 5× fewer iterations than current state-of-the-art “full” DSA, and the proposed method is trivially compatible with voids. On the hohlraum problem, rapid convergence is obtained by preconditioning less than 3% of the mesh elements with five to ten AMG iterations.

A cascadic multigrid asymptotic-preserving discrete ordinate discontinuous streamline diffusion method for radiative transfer equations with diffusive scalings

In this paper, we develop a cascadic multigrid asymptotic-preserving discrete ordinate discontinuous streamline diffusion scheme for radiative transfer equations (RTE) with multiple scalings. Our new method employs a simple and efficient cascadic multigrid method to the discretized RTE system as well as a diffusion synthetic acceleration technique as an efficient smoother to accelerate the convergence of the iteration in diffusive region. Furthermore, by applying the discontinuous streamline diffusion schemes, improved convergence condition in heterogeneous media and the asymptotic-preserving (AP) property can be achieved. The AP property of these methods will be explained formally and demonstrated numerically. Numerical results are presented to show the effectiveness and efficiency of the proposed numerical scheme for solving radiative transfer equations, especially in diffusive and heterogeneous media.

A diffusion synthetic acceleration approach to k-eigenvalue neutron transport using PJFNK

This paper investigates a diffusion synthetic acceleration (DSA) approach to the k-eigenvalue problem of the neutron transport equation. The novelties of this study reside in 1) casting the low-order diffusion equation directly as a nonlinear problem, which is solved with the PJFNK (preconditioned Jacobian-free Newton Krylov) method, circumventing the need for power iteration; 2) showing the appealing features on solving the diffusion equation with DSA compared with nonlinear diffusion acceleration (NDA) with a simplified model of ATR (Advanced Test Reactor); 3) proposing the DSA weak form for the discretization scheme with SAAF (self-adjoint angular flux) formulation that is stable with anisotropic scatterings; 4) showing the impact of the vacuum boundary coefficient of the low-order equation on the convergence of DSA. Both DSA and NDA are implemented in and tested with Rattlesnake code, the INL MOOSE-based radiation transport application for multi-physics modeling and simulations.

Accelerating the solution of the [formula omitted] equations with highly anisotropic scattering using the Fokker-Planck approximation

The discrete ordinates method can model forward-peaked transport problems accurately. However, convergence of discrete ordinates solution can become arbitrarily slow upon use of standard iterative procedures like source iteration. Standard zeroth and first moment-based acceleration methods like nonlinear diffusion acceleration and diffusion synthetic acceleration are ineffective in accelerating such problems because these methods do not correct higher order Legendre-moments of angular flux. In this paper, we explore the idea of using the Fokker-Planck approximation as a preconditioner to accelerate the convergence of solution of forward-peaked transport problems. We observe that such preconditioning can successfully reduce the iteration count by up to four orders of magnitude, and solver runtime by up to three orders of magnitude when compared to diffusion synthetic acceleration.

Thermal Upscattering Acceleration Schemes for Parallel Transport Sweeps

This work is motivated by the need to solve realistic problems with complex energy, space, and angle dependence, which requires parallel multigroup transport sweeps combined with efficient acceleration of the thermal upscattering. We present various iterative schemes based on the two-grid (TG) diffusion synthetic acceleration (DSA) method. In its original form, the TG method is used with the Gauss-Seidel iterative scheme over energy groups, which makes it impractical for parallel computation. We therefore formulate a Jacobi-style version. Furthermore, we propose a new scheme that reduces the overall number of transport sweeps by removing the need to fully converge the within-group iterations before the TG step. This becomes possible by adding an additional within-group DSA solve after each transport sweep. Fourier analyses are carried out to ascertain the effectiveness of the proposed scheme, with further corroboration from massively parallel numerical results from practical problem calculations. We discuss several implementation strategies of the new scheme, paying particular attention to the consequences on the overall efficiency of adding additional diffusion solves with a relatively low number of degrees of freedom per process.

A quadratic programming flux correction method for high-order DG discretizations of SN transport

We present a new flux-fixup approach for arbitrarily high-order discontinuous Galerkin (DG) discretizations of the SN transport equation, and we demonstrate the compatibility of this approach with the Variable Eddington Factor (VEF) method [1,2]. The new fixup approach is sweep-compatible: during a transport sweep (block Gauss-Seidel iteration in which the scattering source is lagged), a local quadratic programming (QP) problem is solved in each spatial element to ensure that the solution satisfies certain physical constraints, including local particle balance. In this paper, we describe two choices of physical constraints, resulting in two variants of the method: QP Zero (QPZ) and QP Maximum Principle (QPMP). In QPZ, the finite element coefficients of the solution are constrained to be nonnegative. In QPMP, they are constrained to adhere to an approximate discrete maximum principle.There are two primary takeaways in this paper. First, when the positive Bernstein basis is used for DG discretization, the QPMP method eliminates negativities, preserves high-order accuracy for smooth problems, and significantly dampens unphysical oscillations in the solution. The latter feature – the dampening of unphysical oscillations – is an improvement upon standard, simpler fixup approaches such as the approach described in [3] (denoted as the “zero and rescale” (ZR) method in this paper). This improvement comes at a moderate computational cost, but it is not prohibitive. Our results show that, even in an unrealistic worst-case scenario where 83% of the spatial elements require a fixup, the computational cost of performing a transport sweep with fixup is only ∼31% greater than performing one without fixup.The second takeaway is that the VEF method can be used to accelerate the convergence of transport sweeps even when a fixup is applied. When optically thick regions are present, transport sweeps converge slowly, regardless of whether a fixup is applied, and acceleration is needed. However, attempting to apply standard diffusion synthetic acceleration (DSA) to fixed-up transport sweeps results in divergence for optically thick problems. Our results show that the same is not true for VEF. When VEF is combined with fixed-up transport sweeps, the result is a scheme that produces a nonnegative solution, converges independently of the mean free path, and, in the case of the QPMP fixup, adheres to an approximate discrete maximum principle.