The Stability of Linear Diffusion Acceleration Relative to CMFD

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.

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DEMONSTRATION OF A LINEAR PROLONGATION CMFD METHOD ON MOC

EPJ Web of Conferences ◽

10.1051/epjconf/202124703006 ◽

2021 ◽

Vol 247 ◽

pp. 03006

Author(s):

Jin Li ◽

Yunlin Xu ◽

Dean Wang ◽

Qicang Shen ◽

Brendan Kochunas ◽

...

Keyword(s):

Neutron Transport ◽

Coarse Mesh ◽

Scalar Flux ◽

Scalar Fluxes ◽

Transport Calculation ◽

Discontinuity Problem ◽

Neutron Transport Equations ◽

The Stability ◽

Partial Current ◽

Better Than

Coarse Mesh Finite Difference (CMFD) method is a very effective method to accelerate the iterations for neutron transport calculation. But it can degrade and even fail when the optical thickness of the mesh becomes large. Therefore several methods, including partial current-based CMFD (pCMFD) and optimally diffusive CMFD (odCMFD), have been proposed to stabilize the conventional CMFD method. Recently, a category of “higherorder” prolongation CMFD (hpCMFD) methods was proposed to use both the local and neighboring coarse mesh fluxes to update the fine cell flux, which can solve the fine cell scalar flux discontinuity problem between the fine cells at the bounary of the coarse mesh. One of the hpCMFD methods, refered as lpCMFD, was proposed to use a linear prolongation to update the fine cell scalar fluxes. Method of Characteristics (MOC) is a very popular method to solve neutron transport equations. In this paper, lpCMFD is applied on the MOC code MPACT for a variety of fine meshes. A track-based centroids calculation method is introduced to find the centroids coordinates for random shapes of fine cells. And the numerical results of a 2D C5G7 problem are provided to demonstrate the stability and efficiency of lpCMFD method on MOC. It shows that lpCMFD can stabilize the CMFD iterations in MOC method effectively and lpCMFD method performs better than odCMFD on reducing the outer MOC iterations.

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A Revisit to CMFD Schemes: Fourier Analysis and Enhancement

Energies ◽

10.3390/en14020424 ◽

2021 ◽

Vol 14 (2) ◽

pp. 424

Author(s):

Dean Wang ◽

Zuolong Zhu

Keyword(s):

Fourier Analysis ◽

Eigenvalue Problems ◽

Neutron Transport ◽

Coarse Mesh ◽

Successive Overrelaxation ◽

Computational Cell ◽

Artificial Diffusion ◽

Acceleration Method ◽

The Stability ◽

Partial Current

The coarse-mesh finite difference (CMFD) scheme is a very effective nonlinear diffusion acceleration method for neutron transport calculations. CMFD can become unstable and fail to converge when the computational cell optical thickness is relatively large in k-eigenvalue problems or diffusive fixed-source problems. Some variants and fixups have been developed to enhance the stability of CMFD, including the partial current-based CMFD (pCMFD), optimally diffusive CMFD (odCMFD), and linear prolongation-based CMFD (lpCMFD). Linearized Fourier analysis has proven to be a very reliable and accurate tool to investigate the convergence rate and stability of such coupled high-order transport/low-order diffusion iterative schemes. It is shown in this paper that the use of different transport solvers in Fourier analysis may have some potential implications on the development of stabilizing techniques, which is exemplified by the odCMFD scheme. A modification to the artificial diffusion coefficients of odCMFD is proposed to improve its stability. In addition, two explicit expressions are presented to calculate local optimal successive overrelaxation (SOR) factors for lpCMFD to further enhance its acceleration performance for fixed-source problems and k-eigenvalue problems, respectively.

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Accuracy contours in (nT, λ) space in electrochemical digital simulations

Collection of Czechoslovak Chemical Communications ◽

10.1135/cccc19910020 ◽

1991 ◽

Vol 56 (1) ◽

pp. 20-41 ◽

Cited By ~ 6

Author(s):

Dieter Britz ◽

Merete F. Nielsen

Keyword(s):

Finite Difference ◽

Linear Sweep Voltammetry ◽

Stability Factor ◽

Dimensionless Time ◽

Simulation Techniques ◽

Rational Algorithm ◽

The Stability ◽

Transport Problems ◽

Current Approximation ◽

Digital Simulations

In finite difference simulations of electrochemical transport problems, it is usually tacitly assumed that λ, the stability factor Dδt/δx2, should be set as high as possible. Here, accuracy contours are shown in (nT, λ) space, where nT is he number of finite difference steps per unit (dimensionless) time. Examples are the Cottrell experiment, simple chronopotentiometry and linear sweep voltammetry (LSV) on a reversible system. The simulation techniques examined include the standard explicit (point- and box-) methods as well as Runge-Kutta, Crank-Nicolson, hopscotch and Saul’yev. For the box method, the two-point current approximation appears to be the most appropriate. A rational algorithm for boundary concentrations with explicit LSV simulations is discussed. In general, the practice of choosing as high a λ value when using the explicit techniques, is confirmed; there are practical limits in all cases.

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An analytical approach for solving a nodal formulation of two-dimensional fixed-source neutron transport problems with linearly anisotropic scattering

Progress in Nuclear Energy ◽

10.1016/j.pnucene.2017.03.017 ◽

2017 ◽

Vol 98 ◽

pp. 193-201 ◽

Cited By ~ 5

Author(s):

C.B. Picoloto ◽

R.D. da Cunha ◽

R.C. Barros ◽

L.B. Barichello

Keyword(s):

Analytical Approach ◽

Neutron Transport ◽

Anisotropic Scattering ◽

Two Dimensional ◽

Source Neutron ◽

Transport Problems

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NUMERICAL SOLUTION OF TRANSIENT AND STEADY-STATE NEUTRON TRANSPORT PROBLEMS

10.2172/4198642 ◽

1959 ◽

Cited By ~ 5

Author(s):

B. Carlson

Keyword(s):

Steady State ◽

Numerical Solution ◽

Neutron Transport ◽

Transport Problems ◽

Transient And Steady State

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The Importance of Anisotropic Scattering in High Energy Neutron Transport Problems

Reactor Dosimetry ◽

10.1007/978-94-009-5378-9_28 ◽

1985 ◽

pp. 287-293

Author(s):

G. Prillinger ◽

M. Mattes

Keyword(s):

Neutron Transport ◽

High Energy ◽

Anisotropic Scattering ◽

High Energy Neutron ◽

Energy Neutron ◽

Transport Problems

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Equations of the surface harmonics method for solving time-dependent neutron transport problems and their verification

Physics of Atomic Nuclei ◽

10.1134/s1063778813140044 ◽

2013 ◽

Vol 76 (14) ◽

pp. 1631-1638 ◽

Cited By ~ 1

Author(s):

V. F. Boyarinov ◽

A. E. Kondrushin ◽

P. A. Fomichenko

Keyword(s):

Neutron Transport ◽

Time Dependent ◽

Transport Problems ◽

Surface Harmonics Method

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Behavior of a Sequence of Geometric Transformations for a Truncated Ellipsoid Geometry in Transport Theory

Volume 5: Fuel Cycle and High and Low Level Waste Management and Decommissioning; Computational Fluid Dynamics (CFD), Neutronics Methods and Coupled Codes; Instrumentation and Control ◽

10.1115/icone17-75758 ◽

2009 ◽

Author(s):

Rube´n Panta Pazos

Keyword(s):

Boundary Conditions ◽

Symmetry Axis ◽

Transport Theory ◽

Neutron Transport ◽

Complex Function ◽

Computational Techniques ◽

Neutron Transport Equation ◽

Geometric Transformations ◽

The Right ◽

Transport Problems

The neutron transport equation has been studied from different approaches, in order to solve different situations. The number of methods and computational techniques has increased recently. In this work we present the behavior of a sequence of geometric transformations evolving different transport problems in order to obtain solve a transport problem in a truncated ellipsoid geometry and subject to known boundary conditions. This scheme was depicted in 8, but now is solved for the different steps. First, it is considered a rectangle domain that consists of three regions, source, void and shield regions 5. Horseshoe domain: for that it is used the complex function: f:D→C,definedasf(z)=12ez+1ezwhereD=z∈C−0.5≤Re(z)≤0.5,−12π≤Im(z)≤12π(0.1) The geometry obtained is such that the source is at the focus of an ellipse, and the target coincides with the other focus. The boundary conditions are reflective in the left boundary and vacuum in the right boundary. Indeed, if the eccentricity is a number between 0,95 and 0,99, the distance between the source and the target ranges from 20 to 100 length units. The rotation around the symmetry axis of the horseshoe domain generates a truncated ellipsoid, such that a focus coincides with the source. In this work it is analyzed the flux in each step, giving numerical results obtained in a computer algebraic system. Applications: in nuclear medicine and others.

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