Implicit discrete ordinates discontinuous Galerkin method for radiation problems on shared-memory multicore CPU/many-core GPU computation architecture

References

• G. N. Lygidakis and I. K. Nikolos, “Assessment of different spatial/angular agglomeration multigrid schemes for the acceleration of FVM radiative heat transfer computations,” Numer. Heat Transfer B, vol. 69, no. 5, pp. 389–412, 2016. DOI: 10.1080/10407790.2016.1138701.
   Web of Science ®Google Scholar
• J. Kophazi and D. Lathouwers, “A space–angle DGFEM approach for the Boltzmann radiation transport equation with local angular refinement,” J. Comput. Phys, vol. 297, pp. 637–668, 2015. DOI: 10.1016/j.jcp.2015.05.031.
   Web of Science ®Google Scholar
• J. S. Warsa, “A continuous finite element-based, discontinuous finite element method for sn transport,” Nucl. Sci. Eng., vol. 160, no. 3, pp. 385–400, 2008. DOI: 10.13182/NSE160-385TN.
   Web of Science ®Google Scholar
• E. Machorro, “Discontinuous Galerkin finite element method applied to the 1-d spherical neutron transport equation,” J. Comput. Phys., vol. 223, no. 1, pp. 67–81, 2007. DOI: 10.1016/j.jcp.2006.08.020.
   Web of Science ®Google Scholar
• S. Giani and M. Seaid, “HP-adaptivity discontinuous Galerkin methods for simplified p_n approximations of frequency-dependent radiative transfer,” Comput. Methods. Appl. Mech. Eng., vol. 301, pp. 52–79, 2016. DOI: 10.1016/j.cma.2015.12.013.
   Web of Science ®Google Scholar
• R. B. Lowrie and J. E. Morel, “Discontinuous Galerkin for hyperbolic systems with stiff relaxation,” in Discontinuous Galerkin Methods: Theory, Computation and Application, B. Cockburn and G. E. Karniadakis, Eds. New York: Springer, 2000, pp. 385–390..
   Google Scholar
• R. McClarren, V. Laboure and C. Hauck, “Implicit filtered Pn for high-energy density thermal radiation transport using discontinuous Galerkin finite elements,” J. Comput. Phys., vol. 321, pp. 624–643, 2016.
   Web of Science ®Google Scholar
• J. Huang, W. Han and J. Eichholz, “Discrete-ordinates discontinuous Galerkin methods for solving the radiative transfer equation,” SIAM J. Sci. Comput, vol. 32, no. 2, pp. 477–497, 2010. DOI: 10.1137/090767340.
   Web of Science ®Google Scholar
• J. L. Guermond and G. Kanschat, “Asymptotic analysis of upwind discontinuous Galerkin approximation of the radiative transport equation in the diffusion limit,” SIAM J. Numer. Anal., vol. 48, no. 1, pp. 53–78, 2010. DOI: 10.1137/090746938.
   Web of Science ®Google Scholar
• Y. Saad, Iterative Methods for Sparse Linear Systems. Boston, MA: International Thomson Publishing, 1996.
   Google Scholar
• Y. Saad and M. H. Schultz, “GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput., vol. 7, no. 3, pp. 856–869, 1986. DOI: 10.1137/0907058.
   Web of Science ®Google Scholar
• H. A. van der Vorst, “BI-CGSTAB: A fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems,” SIAM J. Sci. Stat. Comput., vol. 13, no. 2, pp. 631–644, 1992. DOI: 10.1137/0913035.
   Web of Science ®Google Scholar
• R. Singh and K. M. Singh, “On preconditioned BICGSTAB solver for MLPG method applied to heat conduction in complex geometry,” Numer. Heat Transfer B, vol. 72, no. 5, pp. 377–391, 2017. DOI: 10.1080/10407782.2017.1400335.
   Web of Science ®Google Scholar
• Y. Saad, “Krylov subspace methods on supercomputers,” SIAM J. Sci. Stat. Comput., vol. 10, no. 6, pp. 1200–1232, 1989. DOI: 10.1137/0910073.
   Web of Science ®Google Scholar
• MPI forum. MPI-2: Extensions to the message–passing interfaces. Available: http://www.mpi-forum.org/docs/mpi-20-html/mpi2-report.html, 1997.
   Google Scholar
• OpenMP website. Available: http://www.openmp.org.
   Google Scholar
• J. Tal, R. Ben-Zvil and A. Kribus, “A high-efficiency parallel solution of the radiative transfer equation,” Numer. Heat Transfer B, vol. 44, no. 3, pp. 295–308, 2003. DOI: 10.1080/713836382.
   Web of Science ®Google Scholar
• J. Gon Alves and P. J. Coelho, “Parallelization of the discrete ordinates method,” Numer. Heat Transfer B, vol. 32, no. 2, pp. 151–173, 1997. DOI: 10.1080/10407799708915003.
   Web of Science ®Google Scholar
• Ö. Yιldιz and H. Bedir, “A parallel solution to the radiative transport in three-dimensional participating media,” Numer. Heat Transfer B, vol. 50, no. 1, pp. 79–95, 2006. DOI: 10.1080/10407790500459361.
   Web of Science ®Google Scholar
• G. Krishnamoorthy, R. Rawat and P. J. Smith, “Parallel computations of radiative heat transfer using the discrete ordinates method,” Numer. Heat Transfer B, vol. 47, no. 1, pp. 19–38, 2004. DOI: 10.1080/10407790490487451.
   Web of Science ®Google Scholar
• S. D. Pautz, “An algorithm for parallel SN sweeps on unstructured meshes,” Nucl. Sci. Eng., vol. 140, no. 2, pp. 111–136, 2002. DOI: 10.13182/NSE02-1.
   Web of Science ®Google Scholar
• R. S. Baker and K. R. Koch, “An SN algorithm for the massively parallel cm-200 computer,” Nucl. Sci. Eng., vol. 128, no. 3, pp. 312–320, 1998. DOI: 10.13182/NSE98-1.
   Web of Science ®Google Scholar
• J. Liu, H. M. Shang and Y. S. Chen, “Parallel simulation of radiative heat transfer using an unstructured finite-volume method,” Numer. Heat Transfer B, vol. 36, pp. 115–137, 1999.
   Web of Science ®Google Scholar
• NVIDIA CUDA website. Available: http://developer.nvidia.com/cuda-zone.
   Google Scholar
• W. Fu, W. Wang, C. Li and M. Tsubokura, “An investigation of the unstable phenomena of natural convection in parallel square plates by multi-GPU implementation,” Numer. Heat Transfer B, vol. 71, no. 1, pp. 66–83, 2017. DOI: 10.1080/10407790.2016.1257224.
   Web of Science ®Google Scholar
• M. F. Modest, Radiative Heat Transfer, 2nd ed. New York: Academic Press, 2003.
   Google Scholar
• P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, Computer Science and Applied Mathematics. New York: Academic Press, 1975.
   Google Scholar
• A. Erdelyi and E. Robert, Higher Transcendental Functions, vols. I–III. Huntington, New York: Krieger Publishing Company, 1953.
   Google Scholar
• G. D. Raithby and E. H. Chui, “A finite-volume method for predicting radiant heat transfer in enclosures with participating media,” J. Heat Transfer, vol. 112, no. 2, pp. 415–423, 1990. DOI: 10.1115/1.2910394.
   Web of Science ®Google Scholar
• T. Warburton and J. S. Hesthaven, “Nodal high-order methods on unstructured grids I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys., vol. 181, no. 1, pp. 186–221, 2002. DOI: 10.1006/jcph.2002.7118.
   Web of Science ®Google Scholar
• A. C. Ratzel and J. R. Howell, “Two-dimensional radiation in absorbing–emitting media using the p–n approximation,” J. Heat Transfer, vol. 105, no. 2, pp. 333–340, 1983. DOI: 10.1115/1.3245583.
   Web of Science ®Google Scholar
• Y. Saad and J. Zhang, “BLUTM: A domain-based multilevel block ILUT preconditioner for general sparse matrices,” SIAM J. Matrix Anal. Appl., vol. 21, no. 1, pp. 279–299, 1999. DOI: 10.1137/S0895479898341268.
   Web of Science ®Google Scholar
• Y. Saad, “Multilevel ILU with reorderings for diagonal dominance,” SIAM J. Sci. Comput., vol. 27, no. 3, pp. 1032–1057, 2005. DOI: 10.1137/030602733.
   Web of Science ®Google Scholar
• Y. Saad, “ILUT: A dual threshold incomplete ILU factorization,” Numer. Linear Alg. Appl., vol. 1, no. 4, pp. 387–402, 1994. DOI: 10.1002/nla.1680010405.
   Web of Science ®Google Scholar
• K. Swirydowicz, E. de Sturler, X. Xu and C. J. Roy, “Fast solvers and preconditioners on GPU,” Society for Industrial and Applied Mathematics (SIAM) Annual Meeting (an14), Chicago, Illinois, USA, 2014.
   Google Scholar
• N. Karlsson, “An incompressible Navier–Stokes equations solver on the GPU using CUDA,” Master Thesis, Chalmers University of Technology, Department of Computer Science and Engineering, Goteborg, Sweden, August, 2013.
   Google Scholar