Computational Efficiency of Numerical Methods for the Multigroup, Discrete-Ordinates Neutron Transport Equations: The Slab Geometry Case

Alcouffe, R.E.; Larsen, Edward W.; Miller, W.F.; Wienke, B. R.

doi:10.13182/nse71-111

Cited by 112 publications

(27 citation statements)

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“…The maximum value of |ω DVM |, which occurs for λ = 0, is ω DVM = 1. For a finite system the flat λ = 0 mode, with the corresponding eigenfunction y(µ) = α 0 , cannot be present and thus the DVM described by (1) and (2), when applied to more realistic finite systems with boundary conditions, converges absolutely. At the same time, however, the slow convergence rate of the method in the case of thick slabs is well understood and justified.…”

Section: Stability Analysis Of the Dvmmentioning

confidence: 99%

“…Formulation of the acceleration schemes for the BGK and S models. The development of fast iteration schemes for the solution of the neutron and radiative transport equations has been carried out by many researchers and has led to dramatic theoretical and practical success [1,2,21]. The most advanced of these acceleration methods are based on the formulation of moment equations, the so-called synthetic equations, which are solved coupled with the transport equation to improve significantly the slow iterative convergence rate of the transport equation.…”

Section: Fig 1 Plots Of ω Versus λ For the Typical Dvm (Bgk And S Mmentioning

confidence: 99%

“…This type of fast iterative algorithms has been well developed in the area of neutron transport [11,1,2]. They have been applied effectively to speed up the iterative convergence of the discrete-ordinates method in optically thick regions with low absorption and isotropic or anisotropic scattering [21].…”

Section: Introductionmentioning

confidence: 99%

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Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

Valougeorgis¹,

Naris²

2003

SIAM J. Sci. Comput.

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Abstract. The convergence rate of the discrete velocity method (DVM), which has been applied extensively in the area of rarefied gas dynamics, is studied via a Fourier stability analysis. The spectral radius of the continuum form of the iteration map is found to be equal to one, which justifies the slow convergence rate of the method. Next the efficiency of the DVM is improved by introducing various acceleration schemes. The new synthetic-type schemes speed up significantly the iterative convergence rate. The spectral radius of the acceleration schemes is also studied and the so-called H 1 acceleration method is found to be the optimum one. Finally, the two-dimensional flow problem of a gas through a rectangular microchannel is solved using the new fast iterative DVM. The number of required iterations and the overall computational time are significantly reduced, providing experimental evidence of the analytic formulation. The whole approach is demonstrated using the BGK and S kinetic models.Key words. iterative methods, acceleration schemes, rarefied gas flows AMS subject classifications. 65B99, 76P05PII. S1064827502406506 Introduction. After the early work of Broadwell [4], Huang et al. [10], and Cabannes [6], the discrete velocity method (DVM) has been developed into one of the most common techniques for solving the Boltzmann equation [7,12] and simplified model equations [14,23,15] in the area of rarefied gas dynamics. The method has also been applied to solve mixture problems [20,8]. An extensive review article on internal rarefied gas flows including DVM applications has been given lately by Sharipov [16]. Very recently, new models of discrete velocity gases [5] and mixtures [9] have been introduced indicating that the method can be extended into more general models including polyatomic gases with chemical reactions.The method is based on a discretization of the velocity and space variables by choosing a suitable set of discrete velocities and by applying a consistent finite difference scheme, respectively. Then the collision integral term is approximated by an appropriate quadrature, and the resulting discrete system of equations is solved in an iterative manner. Researchers implementing the DVM are well aware, however, of its slow convergence, particularly when domains with thick subregions are considered [17]. In these cases a large number of iterations are required, and calculations are amenable to accumulated round-off error. Special attention is needed to sustain acceptable accuracy. Even more when multidimensional physical systems are examined, computational effort and time are drastically increased. These types of calculations are now needed to solve in an efficient and accurate manner fluidics applications in micro-electrical-mechanical systems and nanotechnology problems. In these applications, due to the fact that the Navier-Stokes equations are restricted by the hydrodynamic regime, kinetic-type equations and corresponding numerical approaches must

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Section: Stability Analysis Of the Dvmmentioning

confidence: 99%

Section: Fig 1 Plots Of ω Versus λ For the Typical Dvm (Bgk And S Mmentioning

confidence: 99%

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Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

Valougeorgis¹,

Naris²

2003

SIAM J. Sci. Comput.

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“…The index pair iω references a face locally to cell i, while a single index j may be used to reference a face globally. These discretization methods are can be expressed as a family: 19) for each cell i in the interior G, with boundary conditions…”

Section: General Form Of Discretization Methods For the Loqd Equationsmentioning

confidence: 99%

“…The approximation of the low-order equations is based on the linear discontinuous (LD) method that was formulated for the transport equation [18,19,15]. We introduce the spatial mesh…”

Section: Discretizationmentioning

confidence: 99%

Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

Anistratov¹,

Constantinescu²,

Roberts³

et al. 2007

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A Spatial Characteristic Scheme for Multigroup Discrete Ordinates Electron Energy Deposition in Two Dimensions

Datta

Ray

1993

Physica Status Solidi (b)

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Energy deposition calculations are performed due to an incident source of electrons in rectangular slab geometry using the method of discrete ordinates for directions and the multigroup treatment in energy. Using the restricted stopping power formalism to distinguish between large energy loss collisions (catastrophic) and small energy loss collisions (continuous), coupled with a second-order differencing scheme in energy, an improved model of electron energy loss is incorporated in our computational model. Results are presented for isotropic and normal incident sources and compared with available results.

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Computational Efficiency of Numerical Methods for the Multigroup, Discrete-Ordinates Neutron Transport Equations: The Slab Geometry Case

Cited by 112 publications

References 0 publications

Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

Acceleration Schemes of the Discrete Velocity Method: Gaseous Flows in Rectangular Microchannels

Nonlinear Projective-Iteration Methods for Solving Transport Problems on Regular and Unstructured Grids

A Spatial Characteristic Scheme for Multigroup Discrete Ordinates Electron Energy Deposition in Two Dimensions

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