Self-Adaptive Spherical Wavelets for Angular Discretizations of the Boltzmann Transport Equation

Buchan, Andrew G.; Pain, Christopher C.; Eaton, M.D.; Smedley-Stevenson, Richard P.; Goddard, A.J.H.

doi:10.13182/nse08-a2751

Cited by 26 publications

(25 citation statements)

References 16 publications

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“…It has to be noted that contrary to the P N or wavelet based schemes [23][24][25], in our discretization method it is sufficient to deal with the Jacobians of the patches separately. This is due to the use of discontinuous basis functions, leading to block diagonal Jacobian structure, essentially decoupling the patches.…”

Section: Riemann Procedures In Non-orthogonal Basesmentioning

confidence: 99%

“…finite element functions or wavelets [23][24][25][26][27]. We give a brief overview of the most important literature in this area.…”

Section: Introductionmentioning

confidence: 99%

“…Buchan et al combined a wavelet based angular expansion with a continuous spatial discretization [23,25,24] for the first order transport equation. These studies offer a firm theoretical background with respect to the use of hierarchical approximation spaces.…”

Section: Introductionmentioning

confidence: 99%

“…The proposed scheme is based on the first order form of the transport equation discretized using discontinuous finite elements in both space and angle. Spherical triangles on the sphere are used for defining the angular basis functions and are similar to those introduced by Jarrell and Adams and Buchan et al [21][22][23]25,24]. The choice of using discontinuous functions on the sphere eases the refinement and solution operations by decoupling the unknowns from patch to patch.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement

Kópházi

Lathouwers

2015

Journal of Computational Physics

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Section: Riemann Procedures In Non-orthogonal Basesmentioning

confidence: 99%

“…finite element functions or wavelets [23][24][25][26][27]. We give a brief overview of the most important literature in this area.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement

Kópházi

Lathouwers

2015

Journal of Computational Physics

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“…Techniques that resolve these domains efficiently have therefore always taken a central role in this area of research. These include technologies spanning adaptive resolution schemes [34,35,36], multi-scale resolution, acceleration techniques [37] as well as the recasting of the equations into more efficient forms [31,33,38]. However, the vast majority of these technologies are always reliant on the solving of a full scale model.…”

Section: Introductionmentioning

confidence: 99%

A POD reduced‐order model for eigenvalue problems with application to reactor physics

Buchan

Pain

Fang

et al. 2013

Numerical Meth Engineering

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SUMMARYA reduced order model (ROM) based on proper orthogonal decomposition (POD) has been presented and applied to solving eigenvalue problems. The model is constructed via the method of snapshots which is based upon the singular value decomposition of a matrix containing the characteristics of a solution as it evolves through time. Part of the novelty of this work is in how this snapshot data is generated, and this is through the recasting of eigenvalue problem, which is time-independent, into a time dependent form.Instances of time dependent eigenfunction solutions are therefore used to construct the snapshot matrix.The reduced order model's capabilities in efficiently resolving eigenvalue problems that typically become computationally expensive (using standard full model discretisations) has been demonstrated. Whilst the approach can be adapted to most general eigenvalue problems, the examples presented here are based on calculating dominant eigenvalues in reactor physics applications. The approach is shown to reconstruct both the eigenvalues and eigenfunctions accurately using a significantly reduced number of unknowns in comparison to 'full' models based on finite element discretisations. The novelty of this paper therefore includes a new approach to generating snapshots, POD's application to large scale eigenvalue calculations, and ROM's application in reactor physics.

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A sub‐grid scale finite element agglomeration multigrid method with application to the Boltzmann transport equation

Buchan

Pain

Umpleby

et al. 2012

Numerical Meth Engineering

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SUMMARY This article describes a new element agglomeration multigrid method for solving partial differential equations discretised through a sub‐grid scale finite element formulation. The sub‐grid scale discretisation resolves solution variables through their separate coarse and fine scales, and these are mapped between the multigrid levels using a dual set of transfer operators. The sub‐grid scale multigrid method forms coarse linear systems, possessing the same sub‐grid scale structure as the original discretisation, that can be resolved without them being stored in memory. This is necessary for the application of this article in resolving the Boltzmann transport equation as the linear systems become extremely large. The novelty of this article is therefore a matrix‐free multigrid scheme that is integrated within its own sub‐grid scale discretisation using dual transfer operators and applied to the Boltzmann transport equation. The numerical examples presented are designed to show the method's preconditioning capabilities for a Krylov space‐based solver. The problems range in difficulty, geometry and discretisation type, and comparisons made with established methods show this new approach to perform consistently well. Smoothing operators are also analysed and this includes using the generalized minimal residual method. Here, it is shown that an adaptation to the preconditioned Krylov space is necessary for it to work efficiently. Copyright © 2012 John Wiley & Sons, Ltd.

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Self-Adaptive Spherical Wavelets for Angular Discretizations of the Boltzmann Transport Equation

Cited by 26 publications

References 16 publications

A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement

A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement

A POD reduced‐order model for eigenvalue problems with application to reactor physics

A sub‐grid scale finite element agglomeration multigrid method with application to the Boltzmann transport equation

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