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 and 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), 63-71. 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,1990), SIAM, Philadelphia, PA, (1991), 264-271.

[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), 282-293.

[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), 445-463. 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), 9-14. 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), 2362-2388. 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), 1781-1824. 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), 17-51. 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), 867-881. 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), 790-795. doi: 10.1006/jcph.1999.6266.

[11]

M. Shashkov, Conservative Finite Difference Methods, CRC Press, Boca Raton, FL, 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), 1015-1031. 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), 6-10.

[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), 1584-1611. doi: 10.1137/040609215.

[15]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer-Verlag, Berlin Heidelberg, 2005.

[16]

M. L. Wilkins, Computer Simulation of Dynamic Phenomena, Springer-Verlag, Berlin Heidelberg, 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), 8935-8951. 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), 253-275. 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), 475-491.

[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, IEEE Computer Society, (2002), 238-242.

[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), 1352-1368. 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), 995-999. 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), 844-866. 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), 3275-3282. 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), 335-358. doi: 10.1137/S1064827501384755.

show all references

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), 63-71. 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,1990), SIAM, Philadelphia, PA, (1991), 264-271.

[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), 282-293.

[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), 445-463. 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), 9-14. 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), 2362-2388. 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), 1781-1824. 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), 17-51. 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), 867-881. 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), 790-795. doi: 10.1006/jcph.1999.6266.

[11]

M. Shashkov, Conservative Finite Difference Methods, CRC Press, Boca Raton, FL, 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), 1015-1031. 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), 6-10.

[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), 1584-1611. doi: 10.1137/040609215.

[15]

A. Toselli and O. Widlund, Domain Decomposition Methods-Algorithms and Theory, Springer-Verlag, Berlin Heidelberg, 2005.

[16]

M. L. Wilkins, Computer Simulation of Dynamic Phenomena, Springer-Verlag, Berlin Heidelberg, 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), 8935-8951. 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), 253-275. 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), 475-491.

[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, IEEE Computer Society, (2002), 238-242.

[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), 1352-1368. 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), 995-999. 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), 844-866. 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), 3275-3282. 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), 335-358. doi: 10.1137/S1064827501384755.

[1]

Fatemeh Bazikar, Saeed Ketabchi, Hossein Moosaei. Smooth augmented Lagrangian method for twin bounded support vector machine. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021027

[2]

Li Jin, Hongying Huang. Differential equation method based on approximate augmented Lagrangian for nonlinear programming. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2267-2281. doi: 10.3934/jimo.2019053

[3]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks and Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143

[4]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic and Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65

[5]

Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure and Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297

[6]

Jing Xu, Xue-Cheng Tai, Li-Lian Wang. A two-level domain decomposition method for image restoration. Inverse Problems and Imaging, 2010, 4 (3) : 523-545. doi: 10.3934/ipi.2010.4.523

[7]

Mehdi Bastani, Davod Khojasteh Salkuyeh. On the GSOR iteration method for image restoration. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 27-43. doi: 10.3934/naco.2020013

[8]

Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete and Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153

[9]

Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control and Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45

[10]

Ming Huang, Cong Cheng, Yang Li, Zun Quan Xia. The space decomposition method for the sum of nonlinear convex maximum eigenvalues and its applications. Journal of Industrial and Management Optimization, 2020, 16 (4) : 1885-1905. doi: 10.3934/jimo.2019034

[11]

Jagadeesh R. Sonnad, Chetan T. Goudar. Solution of the Michaelis-Menten equation using the decomposition method. Mathematical Biosciences & Engineering, 2009, 6 (1) : 173-188. doi: 10.3934/mbe.2009.6.173

[12]

Xiaomao Deng, Xiao-Chuan Cai, Jun Zou. A parallel space-time domain decomposition method for unsteady source inversion problems. Inverse Problems and Imaging, 2015, 9 (4) : 1069-1091. doi: 10.3934/ipi.2015.9.1069

[13]

José Antonio Carrillo, Yingping Peng, Aneta Wróblewska-Kamińska. Relative entropy method for the relaxation limit of hydrodynamic models. Networks and Heterogeneous Media, 2020, 15 (3) : 369-387. doi: 10.3934/nhm.2020023

[14]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[15]

Yachun Li, Shengguo Zhu. Existence results for compressible radiation hydrodynamic equations with vacuum. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1023-1052. doi: 10.3934/cpaa.2015.14.1023

[16]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465

[17]

Yang Li, Yonghong Ren, Yun Wang, Jian Gu. Convergence analysis of a nonlinear Lagrangian method for nonconvex semidefinite programming with subproblem inexactly solved. Journal of Industrial and Management Optimization, 2015, 11 (1) : 65-81. doi: 10.3934/jimo.2015.11.65

[18]

Yang Li, Liwei Zhang. A nonlinear Lagrangian method based on Log-Sigmoid function for nonconvex semidefinite programming. Journal of Industrial and Management Optimization, 2009, 5 (3) : 651-669. doi: 10.3934/jimo.2009.5.651

[19]

Zhong-Zhi Bai. On convergence of the inner-outer iteration method for computing PageRank. Numerical Algebra, Control and Optimization, 2012, 2 (4) : 855-862. doi: 10.3934/naco.2012.2.855

[20]

Tahereh Salimi Siahkolaei, Davod Khojasteh Salkuyeh. A preconditioned SSOR iteration method for solving complex symmetric system of linear equations. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 483-492. doi: 10.3934/naco.2019033

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (141)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]