Skip to main content
SpringerLink
Log in

Moving mesh finite difference solution of non-equilibrium radiation diffusion equations

  • Original Paper
  • Published:
Numerical Algorithms Aims and scope Submit manuscript

Abstract

A moving mesh finite difference method based on the moving mesh partial differential equation is proposed for the numerical solution of the 2T model for multi-material, non-equilibrium radiation diffusion equations. The model involves nonlinear diffusion coefficients and its solutions stay positive for all time when they are positive initially. Nonlinear diffusion and preservation of solution positivity pose challenges in the numerical solution of the model. A coefficient-freezing predictor-corrector method is used for nonlinear diffusion while a cutoff strategy with a positive threshold is used to keep the solutions positive. Furthermore, a two-level moving mesh strategy and a sparse matrix solver are used to improve the efficiency of the computation. Numerical results for a selection of examples of multi-material non-equilibrium radiation diffusion show that the method is capable of capturing the profiles and local structures of Marshak waves with adequate mesh concentration. The obtained numerical solutions are in good agreement with those in the existing literature. Comparison studies are also made between uniform and adaptive moving meshes and between one-level and two-level moving meshes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. An, H., Jia, X., Walker, H.F.: Anderson acceleration and application to the three-temperature energy equations. J. Comput. Phys. 347, 1–19 (2017)

    MathSciNet  MATH  Google Scholar 

  2. Bowes, R.L., Wilson, J.R.: Numerical Modeling in Applied Physics and Astrophysics. Jones and Bartlett, Boston (1991)

    Google Scholar 

  3. Cash, J.R.: Diagonally implicit Runge-Kutta formulate with error estimate. J. Inst. Math. Appl. 24, 293–301 (1979)

    MATH  Google Scholar 

  4. Castor, J.I.: Radiation Hydrodynamics. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  5. Davis, T.A.: Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Software 30, 196–199 (2004)

    MathSciNet  MATH  Google Scholar 

  6. Dvinsky, A.S.: Adaptive grid generation from harmonic maps on Riemannian manifolds. J. Comput. Phys. 95, 450–476 (1991)

    MathSciNet  MATH  Google Scholar 

  7. Huang, W.: Practical aspects of formulation and solution of moving mesh partial differential equations. J. Comput. Phys. 171, 753–775 (2001)

    MathSciNet  MATH  Google Scholar 

  8. Huang, W.: Variational mesh adaptation:isotropy and equidistribution. J. Comput. Phys. 174, 903–924 (2001)

    MathSciNet  MATH  Google Scholar 

  9. Huang, W., Ren, Y., Russell, R.D.: Moving mesh methods based on moving mesh partial differential equations. J. Comput. Phys. 113, 279–290 (1994)

    MathSciNet  MATH  Google Scholar 

  10. Huang, W., Russell, R.D.: A high dimensional moving mesh strategy. Appl. Numer. Math. 26, 63–76 (1998)

    MathSciNet  MATH  Google Scholar 

  11. Huang, W., Russell, R.D.: Adaptive Moving Mesh Methods. Springer, New York (2011). Applied Mathematical Sciences Series, Vol. 174

    MATH  Google Scholar 

  12. Huang, W., Sun, W.: Variational mesh adaptation II: error estimates and monitor functions. J. Comput. Phys. 184, 619–648 (2003)

    MathSciNet  MATH  Google Scholar 

  13. Knoll, D.A., Chacon, L., Margolin, L.G., Mousseau, V.A.: On balanced approximations for time integration of multiple time scale systems. J. Comput. Phys. 185, 583–611 (2003)

    MATH  Google Scholar 

  14. Knoll, D.A., Lowrie, R.B., Morel, J.E.: Numerical analysis of time integration errors for non-equilibrium radiation diffusion. J. Comput. Phys. 226, 1332–1347 (2007)

    MATH  Google Scholar 

  15. Knoll, D.A., Rider, W.J., Olson, G.L.: An efficient nonlinear solution method for non-equilibrium radiation diffusion. J. Quant. Spect. 63, 15–29 (1999)

    Google Scholar 

  16. Knoll, D.A., Rider, W.J., Olson, G.L.: Nonlinear convergence, accuracy, and time step control in non-equilibrium radiation diffusion. J. Quant. Spect. 70, 25–36 (2001)

    Google Scholar 

  17. Kang, K.S.: P1 Nonconforming finite element method for the solution of radiation transport problems, ICASE Report No. 2002-28

  18. Lapenta, G., Chacón, L.: Cost-effectiveness of fully implicit moving mesh adaptation: a practical investigation in 1D. J. Comput. Phys. 219, 86–103 (2006)

    MATH  Google Scholar 

  19. Lai, X., Sheng, Z., Yuan, G.: Monotone finite volume scheme for three dimensional diffusion equation on tetrahedral meshes. Commun. Comput. Phys. 21, 162–181 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Lowrie, R.B.: A comparison of implicit time integration methods for nonlinear relaxation and diffusion. J. Comput. Phys. 196, 566–590 (2004)

    MATH  Google Scholar 

  21. Lu, C., Huang, W., Van Vleck, E.S.: The cutoff method for numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations. J. Comput. Phys. 242, 24–36 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Li, X., Huang, W.: Anisotropic mesh adaptation for 3D anisotropic diffusion problems with application to fractured reservoir simulation. Numer. Math. Theor. Meth. Appl. 10, 913–940 (2017)

    MathSciNet  MATH  Google Scholar 

  23. Liu, J., Qiu, J., Goman, M., Li, X., Liu, M.: Positivity-preserving Runge-Kutta discontinuous Galerkin method on adaptive Cartesian grid for strong moving shock. Numer. Math. Theor. Meth. Appl. 9, 87–110 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Marshak, R.E.: Effect of radiation on shock wave behavior. Phys. Fluids 1, 24–29 (1958)

    MathSciNet  MATH  Google Scholar 

  25. Mihalas, D., Mahalas, B.W.: Foundations of Radiation Hydrodynamics. Oxford University Press, Oxford (1984)

    MATH  Google Scholar 

  26. Mousseau, V.A., Knoll, D.A., Rider, W.J.: Physical-based preconditioning and the Newton-Krylov method for non-equilibrium radiation diffusion. J. Comput. Phys. 160, 743–765 (2000)

    MATH  Google Scholar 

  27. Mousseau, V.A., Knoll, D.A.: New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion. J. Comput. Phys. 190, 42–51 (2003)

    MATH  Google Scholar 

  28. Ovtchinnikov, S., Cai, X.C.: One-level Newton-Krylov-Schwarz algorithm for unsteady nonlinear radiation diffusion problems. Numer. Lin. Alg. Appl. 11, 867–881 (2004)

    MATH  Google Scholar 

  29. Olson, G.L.: Efficient solution of multi-dimensional flux-limited non-equilibrium radiation diffusion coupled to material conduction with second order time discretization. J. Comput. Phys. 226, 1181–1195 (2007)

    MATH  Google Scholar 

  30. Pomraning, G.C.: The non-equilibrium Marshak wave problem. J. Quant. Spectrosc. Radiat. Transf. 21, 249–261 (1979)

    Google Scholar 

  31. Pernice, M., Philip, B.: Solution of equilibrium radiation diffusion problems using implicit adaptive mesh refinement. SIAM J. Sci. Comput. 27(electronic), 1709–1726 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Philip, B., Wang, Z., Brrill, M.A., Rodriguez, M.R., Pernice, M.: Dynamic implicit 3D adaptive mesh refinement for non-equilibrium radiation diffusion. J. Comput. Phys. 262, 17–37 (2014)

    MathSciNet  Google Scholar 

  33. Rider, W. J., Knoll, D.A., Olson, G.L.: A multigrid Newton-Krylov method for multimaterial equilibrium radiation diffusion. J. Comput. Phys. 152, 164–191 (1999)

    MATH  Google Scholar 

  34. Su, B., Olson, G.L.: Benchmark results for the non-equilibrium Marshak diffusion problem. J. Quant. Spectrosc. Radiat. Transf. 56, 337–351 (1996)

    Google Scholar 

  35. Sheng, Z., Yue, J., Yuan, G.: Monotone finite volume schemes of non-equilibrium radiation diffusion equations on distorted meshes. SIAM J. Sci. Comput. 31, 2915–293 (2009)

    MathSciNet  MATH  Google Scholar 

  36. Spitzer, L., Harm, R.: Transport phenomena in a completely ionized gas. Phys. Rev. 89, 977–981 (1953)

    MATH  Google Scholar 

  37. Wise, E.S., Cox, B.T., Treeby, B.E.: Mesh density functions based on local bandwidth applied to moving mesh methods. Commun. Comput. Phys. 22, 1286–1308 (2017)

    MathSciNet  Google Scholar 

  38. Wu, Y., Wang, H.: An AMG preconditioner for solving the Navier-Stokes equations with a moving mesh finite element method. East. Asia. J. Appl. Math. 6, 353–366 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Yuan, G., Hang, X., Sheng, Z., Yue, J.: Progress in numerical methods for radiation diffusion equations. Chinese J. Comput. Phys. 26, 475–500 (2009)

    Google Scholar 

  40. Yue, J., Yuan, G.: Picard-newton iterative method with time step control for multimaterial non-equilibrium radiation diffusion problem. Commun. Comput. Phys. 10, 844–866 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Yang, X., Huang, W., Qiu, J.: A moving mesh finite difference method for equilibrium radiation diffusion equations. J. Comput. Phys. 298, 661–677 (2015)

    MathSciNet  MATH  Google Scholar 

  42. Zhao, X., Chen, Y., Gao, Y., Yu, C., Li, Y.: Finite volume element methods for non-equilibrium radiation diffusion equations. Int. J. Numer. Meth. Fluids 73, 1059–1080 (2013)

    Google Scholar 

  43. Zhang, H., Zegeling, P.A.: A moving mesh finite difference method for non-monotone solutions of non-equilibrium equations in porous media. Commun. Comput. Phys. 22, 935–964 (2017)

    MathSciNet  Google Scholar 

Download references

Acknowledgments

The work was supported in part by NSFC (China) (Grant No. 11701555), NSAF (China) (Grant No. U1630247), and Science Challenge Project (China) (Grant No. TZ2016002).

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science, China University of Mining Technology, Xuzhou, 221116, Jiangsu, China

    Xiaobo Yang

  2. Department of Mathematics, University of Kansas, Lawrence, KS, 66045, USA

    Weizhang Huang

  3. School of Mathematical Sciences, Xiamen University, Xiamen, 361005, Fujian, China

    Jianxian Qiu

Authors
  1. Xiaobo Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Weizhang Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jianxian Qiu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jianxian Qiu.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yang, X., Huang, W. & Qiu, J. Moving mesh finite difference solution of non-equilibrium radiation diffusion equations. Numer Algor 82, 1409–1440 (2019). https://doi.org/10.1007/s11075-019-00662-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-019-00662-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation