Diffusion Synthetic Acceleration Scheme Research Articles

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.

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.

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.

P-multigrid expansion of hybrid multilevel solvers for discontinuous Galerkin finite element discrete ordinate (DG-FEM-SN) diffusion synthetic acceleration (DSA) of radiation transport algorithms

Effective preconditioning of neutron diffusion problems is necessary for the development of efficient DSA schemes for neutron transport problems. This paper uses P-multigrid techniques to expand two preconditioners designed to solve the MIP diffusion neutron diffusion equation with a discontinuous Galerkin (DG-FEM) framework using first-order elements. These preconditioners are based on projecting the first-order DG-FEM formulation to either a linear continuous or a constant discontinuous FEM system. The P-multigrid expansion allows the preconditioners to be applied to problems discretised with second and higher-order elements. The preconditioning algorithms are defined in the form of both a V-cycle and W-cycle and applied to solve challenging neutron diffusion problems. In addition a hybrid preconditioner using P-multigrid and AMG without a constant or continuous coarsening is used. Their performance is measured against a computationally efficient standard algebraic multigrid preconditioner. The results obtained demonstrate that all preconditioners studied in this paper provide good convergence with the continuous method generally being the most computationally efficient. In terms of memory requirements the preconditioners studied significantly outperform the AMG.

Discontinuous diffusion synthetic acceleration for [formula omitted] transport on 2D arbitrary polygonal meshes

In this paper, a Diffusion Synthetic Acceleration (DSA) technique applied to the Sn radiation transport equation is developed using Piece-Wise Linear Discontinuous (PWLD) finite elements on arbitrary polygonal grids. The discretization of the DSA equations employs an Interior Penalty technique, as is classically done for the stabilization of the diffusion equation using discontinuous finite element approximations. The penalty method yields a system of linear equations that is Symmetric Positive Definite (SPD). Thus, solution techniques such as Preconditioned Conjugate Gradient (PCG) can be effectively employed. Algebraic MultiGrid (AMG) and Symmetric Gauss–Seidel (SGS) are employed as conjugate gradient preconditioners for the DSA system. AMG is shown to be significantly more efficient than SGS. Fourier analyses are carried out and we show that this discontinuous finite element DSA scheme is always stable and effective at reducing the spectral radius for iterative transport solves, even for grids with high-aspect ratio cells. Numerical results are presented for different grid types: quadrilateral, hexagonal, and polygonal grids as well as grids with local mesh adaptivity.

Diffusion Synthetic Acceleration for High-Order Discontinuous Finite Element SN Transport Schemes and Application to Locally Refined Unstructured Meshes

Diffusion synthetic acceleration (DSA) schemes compatible with adaptive mesh refinement (AMR) grids are derived for the SN transport equations discretized using high-order discontinuous finite elements. These schemes are directly obtained from the discretized transport equations by assuming a linear dependence in angle of the angular flux along with an exact Fick’s law and, therefore, are categorized as partially consistent. These schemes are akin to the symmetric interior penalty technique applied to elliptic problems and are all based on a second-order discontinuous finite element discretization of a diffusion equation (as opposed to a mixed or P1 formulation). Therefore, they only have the scalar flux as unknowns. A Fourier analysis has been carried out to determine the convergence properties of the three proposed DSA schemes for various cell optical thicknesses and aspect ratios. Out of the three DSA schemes derived, the modified interior penalty (MIP) scheme is stable and effective for realistic problems, even with distorted elements, but loses effectiveness for some highly heterogeneous configurations. The MIP scheme is also symmetric positive definite and can be solved efficiently with a preconditioned conjugate gradient method. Its implementation in an AMR SN transport code has been performed for both source iteration and GMRes-based transport solves, with polynomial orders up to 4. Numerical results are provided and show good agreement with the Fourier analysis results. Results on AMR grids demonstrate that the cost of DSA can be kept low on locally refined meshes.

A two-mesh adaptive mesh refinement technique for [formula omitted] neutral-particle transport using a higher-order DGFEM

We present a two-mesh Adaptive Mesh Refinement technique for the S N neutral-particle transport equation in 2D. A high-order (up to order 4) Discontinuous Galerkin (DG) finite element discretization is employed with standard upwinding. A suitable Diffusion Synthetic Acceleration (DSA) scheme is derived to precondition the transport solves. The DSA scheme, also based on a DG discretization, is derived directly from the discretized high-order DG S N transport equation. Results are presented to demonstrate the superiority of AMR for DSA-accelerated S N transport solves.

Krylov Iterative Methods and the Degraded Effectiveness of Diffusion Synthetic Acceleration for Multidimensional SN Calculations in Problems with Material Discontinuities

A loss in the effectiveness of diffusion synthetic acceleration (DSA) schemes has been observed with certain SN discretizations on two-dimensional Cartesian grids in the presence of material discontinuities. We will present more evidence supporting the conjecture that DSA effectiveness will degrade for multidimensional problems with discontinuous total cross sections, regardless of the particular physical configuration or spatial discretization. Fourier analysis and numerical experiments help us identify a set of representative problems for which established DSA schemes are ineffective, focusing on diffusive problems for which DSA is most needed. We consider a lumped, linear discontinuous spatial discretization of the SN transport equation on three-dimensional, unstructured tetrahedral meshes and look at a fully consistent and a “partially consistent” DSA method for this discretization. The effectiveness of both methods is shown to degrade significantly. A Fourier analysis of the fully consistent DSA scheme in the limit of decreasing cell optical thickness supports the view that the DSA itself is failing when material discontinuities are present in a problem. We show that a Krylov iterative method, preconditioned with DSA, is an effective remedy that can be used to efficiently compute solutions for this class of problems. We show that as a preconditioner to the Krylov method, a partially consistent DSA method is more than adequate. In fact, it is preferable to a fully consistent method because the partially consistent method is based on a continuous finite element discretization of the diffusion equation that can be solved relatively easily. The Krylov method can be implemented in terms of the original SN source iteration coding with only slight modification. Results from numerical experiments show that replacing source iteration with a preconditioned Krylov method can efficiently solve problems that are virtually intractable with accelerated source iteration.

Fully Consistent Diffusion Synthetic Acceleration of Linear Discontinuous SN Transport Discretizations on Unstructured Tetrahedral Meshes

We recently presented a method for efficiently solving linear discontinuous discretizations of the two-dimensional P1 equations on rectangular meshes. The linear system was efficiently solved with Krylov iterative methods and a novel two-level preconditioner based on a linear continuous finite element discretization of the diffusion equation. Here, we extend the preconditioned solution method to three-dimensional, unstructured tetrahedral meshes. Solution of the P1 equations forms the basis of a diffusion synthetic acceleration (DSA) scheme for three-dimensional SN transport calculations with isotropic scattering. The P1 equations and the transport equation are both discretized with isoparametric linear discontinuous finite elements so that the DSA method is fully consistent. Fourier analysis in three dimensions and computational results show that this DSA scheme is stable and very effective. The fully consistent method is compared to other “partially consistent” DSA schemes. Results show that the effectiveness of the partially consistent schemes can degrade for skewed or optically thick mesh cells. In fact, one such scheme can degrade to the extent of being unstable even though it is both unconditionally stable and effective on rectangular grids. Results for a model application show that our fully consistent DSA method can outperform the partially consistent DSA schemes under certain circumstances.

The Even-Parity and Simplified Even-Parity Transport Equations in Two-Dimensionalx-yGeometry

The finite element and lumped finite element methods for the spatial differencing of the even-parity discrete ordinates neutron transport equations (EPS{sub N}) in two-dimensional x-y geometry are applied. In addition, the simplified even-parity discrete ordinates equations (SEPS{sub N}) as an approximation to the EPS{sub N} transport equations are developed. The SEPS{sub N} equations are more efficient to solve than the EPS{sub N} equations due to a reduction in angular domain of one-half, the applicability of a simple five-point diffusion operator, and directionally uncoupled reflective boundary conditions. Furthermore, the SEPS{sub N} equations satisfy the same diffusion limits as EPS{sub N} in an optically thick regime, appear to have no ray effect, and converge faster than EPS{sub N} when using a diffusion synthetic acceleration (DSA). Also, unlike the case of EPS{sub N}, the SEPS{sub N} solutions are strictly positive, thus requiring no negative flux fixups. It is also demonstrated that SEPS{sub N} is a generalization of the simplified P{sub N} method. Most importantly, in these second-order approaches, an unconditionally effective DSA scheme can be achieved by simply integrating the differenced EPS{sub N} and SEPS{sub N} equations over the angles. It is difficult to obtain a consistent DSA scheme with the first-order S{submore » N} equations. This is because a second-order DSA equation must generally be derived directly from the differenced first-order S{sub N} equations.« less

A Diffusion-Synthetic Acceleration Technique for the Even-ParitySnEquations with Anisotropic Scattering

A source iteration scheme and associated diffusion-synthetic acceleration scheme are defined for the even-parity Sn equations with anisotropic scattering. The spatially analytic versions of these s...

Diffusion-Accelerated Solution of the Two-DimensionalX-Y SnEquations with Linear-Bilinear Nodal Differencing

Recently, a new diffusion synthetic acceleration scheme was developed for solving the two-dimensional S[sub n] equations in x-y geometry with bilinear-discontinuous finite element spatial discretization, by using a bilinear-discontinuous diffusion differencing scheme for the diffusion acceleration equations. This method differed from previous methods in that it is unconditionally efficient for problems with isotropic or nearly isotropic scattering. Here, the same bilinear-discontinuous diffusion differencing scheme, and associated multilevel solution technique, is used to accelerate the x-y geometry S[sub n] equations with linear-bilinear nodal spatial differencing. It is found that for problems with isotropic or nearly isotropic scattering, this leads to an unconditionally efficient solution method. Computational results are given that demonstrate this property.

Diffusion-Accelerated Solution of the Two-DimensionalSnEquations with Bilinear-Discontinuous Differencing

A new diffusion synthetic acceleration scheme is developed for solving the two-dimensional S[sub n] equations in x - y geometry with bilinear-discontinuous finite element spatial discretization. This method differs from previous methods in that it is unconditionally efficient for problems with isotropic or weakly anisotropic scattering. Computational results are given that demonstrate this property.