High-Order Finite Element Second Moment Methods for Linear Transport

References

• E. LEWIS and W. MILLER Jr., “A Comparison of P1 Synthetic Acceleration Techniques,” Trans. Am. Nucl. Soc., 23, 202 (1976).
   Google Scholar
• M. L. ADAMS and E. W. LARSEN, “Fast Iterative Methods for Discrete-Ordinates Particle Transport Calculations,” Prog. Nucl. Energy, 40, 1, 3 (2002); https://www.sciencedirect.com/science/article/pii/S0149197001000233.
   Web of Science ®Google Scholar
• L. CHACÓN et al., “Multiscale High-Order/Low-Order (HOLO) Algorithms and Applications,” J. Comput. Phys., 330, 21 (2017); https://doi.org/10.1016/j.jcp.2016.10.069.
   Web of Science ®Google Scholar
• J. WARSA and D. ANISTRATOV, “Two-Level Transport Methods with Independent Discretization,” J. Comput. Theor. Transport, 47, 4–6, 424 (2018); http://dx.doi.org/10.1080/23324309.2018.1497991.
   Web of Science ®Google Scholar
• D. Y. ANISTRATOV and V. Y. GOL’DIN, “Nonlinear Methods for Solving Particle Transport Problems,” Transp. Theory Stat. Phys., 22, 2–3, 125 (1993); http://dx.doi.org/10.1080/00411459308203810.
   Google Scholar
• D. MIHALAS, Stellar Atmospheres, W. H. Freeman and Company (1978).
   Google Scholar
• V. Y. GOL’DIN, “A Quasi-Diffusion Method of Solving the Kinetic Equation,” USSR Comput. Math. Math. Phys., 4, 136 (1964).
   Google Scholar
• S. OLIVIER et al., “A Family of Independent Variable Eddington Factor Methods with Efficient Preconditioned Iterative Solvers,” J. Comput. Phys., 473, 111747 (2023); https://www.sciencedirect.com/science/article/pii/S0021999122008105.
   Web of Science ®Google Scholar
• G. R. CEFUS and E. W. LARSEN, “Stability Analysis of the Quasideffusion and Second Moment Methods for Iteratively Solving Discrete-Ordinates Problems,” Transp. Theory Stat. Phys., 18, 5–6, 493 (1989); http://dx.doi.org/10.1080/00411458908204700.
   Google Scholar
• S. OLIVIER and T. S. HAUT, “High-Order Mixed Finite Element Variable Eddington Factor Methods” (2023); https://arxiv.org/abs/2301.04758.
   Google Scholar
• V. A. DOBREV, T. V. KOLEV, and R. N. RIEBEN, “High-Order Curvilinear Finite Element Methods for Lagrangian Hydrodynamics,” SIAM J. Sci. Comput., 34, 5, B606 (2012); https://doi.org/10.1137/120864672.
   Web of Science ®Google Scholar
• P. G. MAGINOT and T. A. BRUNNER, “Lumping Techniques for Mixed Finite Element Diffusion Discretizations,” J. Comput. Theor. Transport, 47, 4–6, 301 (2018); http://dx.doi.org/10.1080/23324309.2018.1461653.
   Web of Science ®Google Scholar
• V. DOBREV et al., “Algebraic Hybridization and Static Condensation with Application to Scalable H(div) Preconditioning,” SIAM J. Sci. Comput., 41, 3, B425 (2019); http://dx.doi.org/10.1137/17M1132562.
   Web of Science ®Google Scholar
• J. LAUTARD and F. MOREAU, “A Fast 3-d Parallel Diffusion Solver Based on a Mixed-Dual Finite Element Approximation,” Proc. Topl. Mtg. Mathematical Methods and Supercomputing in Nuclear Applications, Karlsruhe, Germany, 1993, American Nuclear Society (1993).
   Google Scholar
• J. P. HENNART, E. H. MUND, and E. D. VALLE, “A Composite Nodal Finite Element for Hexagons,” Nucl. Sci. Eng., 127, 2, 139 (1997); http://dx.doi.org/10.13182/NSE97-A28593.
   Web of Science ®Google Scholar
• A.-M. BAUDRON and J.-J. LAUTARD, “Minos: A Simplified Pn Solver for Core Calculation,” Nucl. Sci. Eng., 155, 2, 250 (2007); http://dx.doi.org/10.13182/NSE07-A2660.
   Web of Science ®Google Scholar
• R. ALCOUFFE, “Diffusion Synthetic Acceleration Methods for the Diamond-Differenced Discrete-Ordinates Equations,” Nucl. Sci. Eng., 64, 344 (1977); http://dx.doi.org/10.13182/NSE77-1.
   Web of Science ®Google Scholar
• J. S. WARSA, T. A. WAREING, and J. E. MOREL, “Fully Consistent Diffusion Synthetic Acceleration of Linear Discontinuous SN Transport Discretizations on Unstructured Tetrahedral Meshes,” Nucl. Sci. Eng., 141, 3, 236 (2002); http://dx.doi.org/10.13182/NSE141-236.
   Web of Science ®Google Scholar
• M. L. ADAMS and W. R. MARTIN, “Diffusion Synthetic Acceleration of Discontinuous Finite Element Transport Iterations,” Nucl. Sci. Eng., 111, 2, 145 (1992); http://dx.doi.org/10.13182/NSE92-A23930.
   Web of Science ®Google Scholar
• Y. WANG and J. C. RAGUSA, “Diffusion Synthetic Acceleration for High-Order Discontinuous Finite Element Sn Transport Schemes and Application to Locally Refined Unstructured Meshes,” Nucl. Sci. Eng., 166, 2, 145 (2010); http://dx.doi.org/10.13182/NSE09-46.
   Web of Science ®Google Scholar
• T. S. HAUT et al., “Diffusion Synthetic Acceleration Preconditioning for Discontinuous Galerkin Discretizations of SN Transport on High-Order Curved Meshes,” SIAM J. Sci. Comput., 42, 5, B1271 (2020); http://dx.doi.org/10.1137/19M124993X.
   Web of Science ®Google Scholar
• N. D. STEHLE, D. Y. ANISTRATOV, and M. L. ADAMS, “A Hybrid Transport-Diffusion Method for 2D Transport Problems with Diffusive Subdomains,” J. Comput. Phys., 270, 325 (2014); http://dx.doi.org/10.1016/j.jcp.2014.03.056.
   Web of Science ®Google Scholar
• P. G. MAGINOT, P. F. NOWAK, and M. L. ADAMS, “A Review of the Upstream Corner Balance Spatial Discretization,” Proc. Int. Conf. Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2017), Jeju, Korea, 2017.
   Google Scholar
• D. Y. ANISTRATOV et al., “Multilevel Second-Moment Methods with Group Decomposition for Multigroup Transport Problems,” Proc. Int. Conf. Mathematics and Computational Methods Applied to Nuclear Science and Engineering (M&C 2021), Virtual Meeting, October 3–7, 2021, American Nuclear Society (2021).
   Google Scholar
• D. Y. ANISTRATOV and J. S. WARSA, “Discontinuous Finite Element Quasi-Diffusion Methods,” Nucl. Sci. Eng., 191, 2, 105 (2018); http://dx.doi.org/10.1080/00295639.2018.1450013.
   Web of Science ®Google Scholar
• D. G. ANDERSON, “Iterative Procedures for Nonlinear Integral Equations,” J. ACM, 12, 4, 547 (Oct 1965); http://dx.doi.org/10.1145/321296.321305.
   Web of Science ®Google Scholar
• M. VAĬNBERG, Variational Methods for the Study of Nonlinear Operators, with a Chapter on Newton’s Method by L. V. KANTOROVICH and G. P. AKILOV, Translated and Supplemented by A. FEINSTEIN, Holden-Day Series in Mathematical Physics, San Francisco, California (1964).
   Google Scholar
• P. CIARLET, Three-Dimensional Elasticity, in Mathematical Elasticity, Elsevier Science (1988).
   Google Scholar
• M. E. ROGNES, R. C. KIRBY, and A. LOGG, “Efficient Assembly of H (div) and H (curl) Conforming Finite Elements,” SIAM J. Sci. Comput., 31, 6, 4130 (2010); http://dx.doi.org/10.1137/08073901X.
   Web of Science ®Google Scholar
• A. QUARTERONI and A. VALLI, Numerical Approximation of Partial Differential Equations, Springer, Berlin, Heidelberg (1994).
   Google Scholar
• P. A. RAVIART and J. M. THOMAS, “A Mixed Finite Element Method for 2-nd Order Elliptic Problems,” Lecture Notes in Mathematics, pp. 292–315, Springer Berlin Heidelberg (1977).
   Google Scholar
• P. RAVIART and J. THOMAS, “Dual Finite Element Models for Second Order Elliptic Problems,” Energy Methods in Finite Element Analysis, p. 175.
   Google Scholar
• D. WOODS, “Discrete Ordinates Radiation Transport Using High-Order Finite Element Spatial Discretizations on Meshes with Curved Surfaces,” PhD Dissertation, Oregon State University (2018).
   Google Scholar
• T. S. HAUT et al., “An Efficient Sweep-Based Solver for the SN Equations on High-Order Meshes,” Nucl. Sci. Eng., 193, 746 (2019); http://dx.doi.org/10.1080/00295639.2018.1562778.
   Web of Science ®Google Scholar
• D. N. ARNOLD, “An Interior Penalty Finite Element Method with Discontinuous Elements,” SIAM J. Numer. Anal., 19, 4, 742 (Aug. 1982); http://dx.doi.org/10.1137/0719052.
   Web of Science ®Google Scholar
• D. N. ARNOLD et al., “Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems,” SIAM J. Numer. Anal., 39, 5, 1749 (2002); http://dx.doi.org/10.1137/S0036142901384162.
   Web of Science ®Google Scholar
• W. PAZNER and T. KOLEV, “Uniform Subspace Correction Preconditioners for Discontinuous Galerkin Methods with hp-Refinement,” Communications on Applied Mathematics and Computation, Lawrence Livermore National Laboratory (July 2021).
   Google Scholar
• F. BASSI and S. REBAY, “A High Order Discontinuous Galerkin Method for Compressible Turbulent Flows,” Discontinuous Galerkin Methods, pp. 77–88, B. COCKBURN, G. E. KARNIADAKIS, and C.-W. SHU, Eds., Springer, Berlin Heidelberg (2000).
   Google Scholar
• B. COCKBURN and C.-W. SHU, “The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems,” SIAM J. Numer. Anal., 35, 6, 2440 (Dec. 1998); http://dx.doi.org/10.1137/S0036142997316712.
   Web of Science ®Google Scholar
• R. ANDERSON et al., “MFEM: A Modular Finite Element Methods Library,” Computers and Mathematics with Applications (July 2020).
   Google Scholar
• R. D. FALGOUT and U. M. YANG, “Hypre: A Library of High Performance Preconditioners,” Proc. Int. Conf. Computational Science (ICCS 2002), Amsterdam, The Netherlands, April 21–24, 2002, Part III, p. 632, Springer-Verlag, Berlin, Heidelberg (2002).
   Google Scholar
• A. C. HINDMARSH et al., “SUNDIALS: Suite of Nonlinear and Differential/Algebraic Equation Solvers,” ACM Trans. Math. Software, 31, 3, 363 (2005); http://dx.doi.org/10.1145/1089014.1089020.
   Web of Science ®Google Scholar
• F. BASSI and S. REBAY, “A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier–Stokes Equations,” J. Comput. Phys., 131, 2, 267 (1997); http://dx.doi.org/10.1006/jcph.1996.5572.
   Web of Science ®Google Scholar
• M. BENZI, G. H. GOLUB, and J. LIESEN, “Numerical Solution of Saddle Point Problems,” Acta Numer., 14, 1 (2005); http://dx.doi.org/10.1017/S0962492904000212.
   Google Scholar
• S. HAMILTON, M. BENZI, and J. WARSA, “Negative Flux Fixups in Discontinuous Finite Element SN Transport,” Proc. Int. Conf. Mathematics, Computational Methods and Reactor Physics (M&C 2009), Saratoga Springs, New York, May 3–7, 2009, American Nuclear Society (2009).
   Google Scholar