# American Institute of Mathematical Sciences

July  2016, 21(5): 1651-1669. doi: 10.3934/dcdsb.2016016

## Parallelization methods for solving three-temperature radiation-hydrodynamic problems

 1 Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China, China

Received  November 2013 Revised  March 2014 Published  April 2016

An efficient parallelization method for numerically solving Lagrangian radiation hydrodynamic problems with three-temperature modeling on structural quadrilateral grids is presented. The three-temperature heat conduction equations are discretized by implicit scheme, and their computational cost are very expensive. Thus a parallel iterative method for three-temperature system of equations is constructed, which is based on domain decomposition for physical space, and combined with fixed point (Picard) nonlinear iteration to solve sub-domain problems. It can avoid global communication and can be naturally implemented on massive parallel computers. The space discretization of heat conduction equations uses the well-known local support operator method (LSOM). Numerical experiments show that the parallel iterative method preserves the same accuracy as the fully implicit scheme, and has high parallel efficiency and good stability, so it provides an effective solution procedure for numerical simulation of the radiation hydrodynamic problems on parallel computers.
Citation: Guangwei Yuan, Yanzhong Yao. Parallelization methods for solving three-temperature radiation-hydrodynamic problems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1651-1669. doi: 10.3934/dcdsb.2016016
##### References:
 [1] C. N. Dawson, Q. Du and T. F. Dupont, A finite difference domain decomposition algorithm for numerical solution of the heat equation,, Math. Comp., 57 (1991), 63. doi: 10.1090/S0025-5718-1991-1079011-4. [2] M. Dryja, Substructuring methods for parabolic problems,, in Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations(Moscow, (1991), 264. [3] S. Gunter and K. Lackner, A mixed implicit-explicit finite difference scheme for heat transport in magnetized plasmas,, Journal of Computational Physics, 228 (2009), 282. [4] E. Jamelot and P. C. Jr, Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation,, Journal of Computational Physics, 241 (2013), 445. doi: 10.1016/j.jcp.2013.01.026. [5] Yu. M. Laevsky and O. V. Rudenko, Splitting methods for parabolic problems in nonrectangular domains,, Appl. Math. Lett., 8 (1995), 9. doi: 10.1016/0893-9659(95)00077-4. [6] H. L. Liao, H. S. Shi and Z. Z. Sun, Corrected explicit-implicit domain decomposition algorithms for two-dimensional semilinear parabolic equations,, Science in China Series A: Mathematics, 52 (2009), 2362. doi: 10.1007/s11425-009-0040-8. [7] P. H. Maire, R. Abgrall, J. Breil and J. Ovadia, A centered Lagrangian scheme for multidimensional compressible flow problems,, SIAM Journal on Scientific Computing, 29 (2007), 1781. doi: 10.1137/050633019. [8] J. E. Morel, R. M. Roberts and M. J. Shashkov, A local support-operators diffusion discretization scheme for quadrilateral $r-z$ meshes,, Journal of Computational Physics, 144 (1998), 17. doi: 10.1006/jcph.1998.5981. [9] S. Ovtchinnikov and X. C. Cai, One-level Newton-Krylov-Schwarz algorithm for unsteady non-linear radiation diffusion problem,, Numerical Linear Algebra with Applications, 11 (2004), 867. doi: 10.1002/nla.386. [10] W. J. Rider and D. A. Knoll, Time step size selection for radiation diffusion calculations,, Journal of Computational Physics, 152 (1999), 790. doi: 10.1006/jcph.1999.6266. [11] M. Shashkov, Conservative Finite Difference Methods,, CRC Press, (1996). [12] Z. Q. Sheng, G. W. Yuan and X. D. Hang, Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation,, Applied Mathematics and Computation, 184 (2007), 1015. doi: 10.1016/j.amc.2006.07.003. [13] A. Shestakov, J. Milovich and D. Kershaw, Parallelization of an unstructured-grid, laser fusion design code,, SIAM News, 32 (1999), 6. [14] H. S. Shi and H. L. Liao, Unconditional stability of corrected explicit-implicit domain decomposition algorithms for parallel approximation of heat equations,, SIAM J. Numer. Anal., 44 (2006), 1584. doi: 10.1137/040609215. [15] A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory,, Springer-Verlag, (2005). [16] M. L. Wilkins, Computer Simulation of Dynamic Phenomena,, Springer-Verlag, (1999). doi: 10.1007/978-3-662-03885-7. [17] L. Yin, J. M. Wu and Y. Z. Yao, A cell functional minimization scheme for parabolic problem,, Journal of Computational Physics, 229 (2010), 8935. doi: 10.1016/j.jcp.2010.08.018. [18] G. W. Yuan, X. D. Hang and Z. Q. Sheng, Parallel difference schemes with interface extrapolation terms for quasi-linear parabolic systems,, Science in China Series A: Mathematics, 50 (2007), 253. doi: 10.1007/s11425-007-0014-7. [19] G. W. Yuan and X. D. Hang, Conservative parallel schemes for diffusion equations,, Chinese Journal of Computational Physics, 27 (2010), 475. [20] G. W. Yuan, L. G. Shen and Y. L. Zhou, Parallel Difference Schemes for Parabolic Problem,, in Proceeding of 2002 5th International Conference on Algorithms and Architectures for Parallel Processing, (2002), 238. [21] G. W. Yuan, Y. Z. Yao and L. Yin, Conservative domain decomposition procedure for nonlinear diffusion problems on arbitrary quadrilateral grids,, SIAM J. Sci. Comput., 33 (2011), 1352. doi: 10.1137/10081335X. [22] G. W. Yuan and F. L. Zuo, Parallel differences schemes for heat conduction equations,, International Journal of Computer Mathematics, 80 (2003), 995. doi: 10.1080/0020716031000087159. [23] J. Y. Yue and G. W. Yuan, Picard-Newton iterative method with time step control for multimaterial non-equilibrium radiation diffusion problem,, Commun. Comput. Phys., 10 (2011), 844. doi: 10.4208/cicp.310110.161010a. [24] S. H. Zhu, Conservative domain decomposition procedure with unconditional stability and second-order accuracy,, Applied Mathematics and Computation, 216 (2010), 3275. doi: 10.1016/j.amc.2010.04.054. [25] Y. Zhuang and X. Sun, Stabilized explicit-implicit domain decomposition methods for the numerical solution of parabolic equations,, SIAM J. Sci. Comput., 24 (2002), 335. doi: 10.1137/S1064827501384755.

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##### References:
 [1] C. N. Dawson, Q. Du and T. F. Dupont, A finite difference domain decomposition algorithm for numerical solution of the heat equation,, Math. Comp., 57 (1991), 63. doi: 10.1090/S0025-5718-1991-1079011-4. [2] M. Dryja, Substructuring methods for parabolic problems,, in Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations(Moscow, (1991), 264. [3] S. Gunter and K. Lackner, A mixed implicit-explicit finite difference scheme for heat transport in magnetized plasmas,, Journal of Computational Physics, 228 (2009), 282. [4] E. Jamelot and P. C. Jr, Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation,, Journal of Computational Physics, 241 (2013), 445. doi: 10.1016/j.jcp.2013.01.026. [5] Yu. M. Laevsky and O. V. Rudenko, Splitting methods for parabolic problems in nonrectangular domains,, Appl. Math. Lett., 8 (1995), 9. doi: 10.1016/0893-9659(95)00077-4. [6] H. L. Liao, H. S. Shi and Z. Z. Sun, Corrected explicit-implicit domain decomposition algorithms for two-dimensional semilinear parabolic equations,, Science in China Series A: Mathematics, 52 (2009), 2362. doi: 10.1007/s11425-009-0040-8. [7] P. H. Maire, R. Abgrall, J. Breil and J. Ovadia, A centered Lagrangian scheme for multidimensional compressible flow problems,, SIAM Journal on Scientific Computing, 29 (2007), 1781. doi: 10.1137/050633019. [8] J. E. Morel, R. M. Roberts and M. J. Shashkov, A local support-operators diffusion discretization scheme for quadrilateral $r-z$ meshes,, Journal of Computational Physics, 144 (1998), 17. doi: 10.1006/jcph.1998.5981. [9] S. Ovtchinnikov and X. C. Cai, One-level Newton-Krylov-Schwarz algorithm for unsteady non-linear radiation diffusion problem,, Numerical Linear Algebra with Applications, 11 (2004), 867. doi: 10.1002/nla.386. [10] W. J. Rider and D. A. Knoll, Time step size selection for radiation diffusion calculations,, Journal of Computational Physics, 152 (1999), 790. doi: 10.1006/jcph.1999.6266. [11] M. Shashkov, Conservative Finite Difference Methods,, CRC Press, (1996). [12] Z. Q. Sheng, G. W. Yuan and X. D. Hang, Unconditional stability of parallel difference schemes with second order accuracy for parabolic equation,, Applied Mathematics and Computation, 184 (2007), 1015. doi: 10.1016/j.amc.2006.07.003. [13] A. Shestakov, J. Milovich and D. Kershaw, Parallelization of an unstructured-grid, laser fusion design code,, SIAM News, 32 (1999), 6. [14] H. S. Shi and H. L. Liao, Unconditional stability of corrected explicit-implicit domain decomposition algorithms for parallel approximation of heat equations,, SIAM J. Numer. Anal., 44 (2006), 1584. doi: 10.1137/040609215. [15] A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory,, Springer-Verlag, (2005). [16] M. L. Wilkins, Computer Simulation of Dynamic Phenomena,, Springer-Verlag, (1999). doi: 10.1007/978-3-662-03885-7. [17] L. Yin, J. M. Wu and Y. Z. Yao, A cell functional minimization scheme for parabolic problem,, Journal of Computational Physics, 229 (2010), 8935. doi: 10.1016/j.jcp.2010.08.018. [18] G. W. Yuan, X. D. Hang and Z. Q. Sheng, Parallel difference schemes with interface extrapolation terms for quasi-linear parabolic systems,, Science in China Series A: Mathematics, 50 (2007), 253. doi: 10.1007/s11425-007-0014-7. [19] G. W. Yuan and X. D. Hang, Conservative parallel schemes for diffusion equations,, Chinese Journal of Computational Physics, 27 (2010), 475. [20] G. W. Yuan, L. G. Shen and Y. L. Zhou, Parallel Difference Schemes for Parabolic Problem,, in Proceeding of 2002 5th International Conference on Algorithms and Architectures for Parallel Processing, (2002), 238. [21] G. W. Yuan, Y. Z. Yao and L. Yin, Conservative domain decomposition procedure for nonlinear diffusion problems on arbitrary quadrilateral grids,, SIAM J. Sci. Comput., 33 (2011), 1352. doi: 10.1137/10081335X. [22] G. W. Yuan and F. L. Zuo, Parallel differences schemes for heat conduction equations,, International Journal of Computer Mathematics, 80 (2003), 995. doi: 10.1080/0020716031000087159. [23] J. Y. Yue and G. W. Yuan, Picard-Newton iterative method with time step control for multimaterial non-equilibrium radiation diffusion problem,, Commun. Comput. Phys., 10 (2011), 844. doi: 10.4208/cicp.310110.161010a. [24] S. H. Zhu, Conservative domain decomposition procedure with unconditional stability and second-order accuracy,, Applied Mathematics and Computation, 216 (2010), 3275. doi: 10.1016/j.amc.2010.04.054. [25] Y. Zhuang and X. Sun, Stabilized explicit-implicit domain decomposition methods for the numerical solution of parabolic equations,, SIAM J. Sci. Comput., 24 (2002), 335. doi: 10.1137/S1064827501384755.
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