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Finite Volume Element Methods for Two-Dimensional Three-Temperature Radiation Diffusion Equations

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  20 July 2016

Yanni Gao ,
Xiukun Zhao  and
Yonghai Li
Yanni Gao*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, Jilin, People's Republic of China
Xiukun Zhao*
Affiliation:
Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600MB Eindhoven, The Netherlands
Yonghai Li*
Affiliation:
School of Mathematics, Jilin University, Changchun 130012, Jilin, People's Republic of China
*
*Corresponding author. Email addresses:yonghai@jlu.edu.cn (Y.-H. Li), gaoyn10@163.com (Y.-N. Gao), ying509501@126.com (X.-K. Zhao)
*Corresponding author. Email addresses:yonghai@jlu.edu.cn (Y.-H. Li), gaoyn10@163.com (Y.-N. Gao), ying509501@126.com (X.-K. Zhao)
*Corresponding author. Email addresses:yonghai@jlu.edu.cn (Y.-H. Li), gaoyn10@163.com (Y.-N. Gao), ying509501@126.com (X.-K. Zhao)
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Abstract

Two-dimensional three-temperature (2-D 3-T) radiation diffusion equations are widely used to approximately describe the evolution of radiation energy within a multi-material system and explain the exchange of energy among electrons, ions and photons. Their highly nonlinear, strong discontinuous and tightly coupled phenomena always make the numerical solution of such equations extremely challenging. In this paper, we construct two finite volume element schemes both satisfying the discrete conservation property. One of them can well preserve the positivity of analytical solutions, while the other one does not satisfy this property. To fix this defect, two as repair techniques are designed. In addition, as the numerical simulation of 2-D 3-T equations is very time consuming, we also devise a mesh adaptation algorithm to reduce the cost. Numerical results show that these new methods are practical and efficient in solving this kind of problems.

Keywords

Two-dimensional three-temperature radiation diffusion equationsFinite volume element methodEnergy conservation propertyRepair techniqueMesh adaptationCutoff method

MSC classification

Secondary: 65N08: Finite volume methods 65N12: Stability and convergence of numerical methods 65N15: Error bounds
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 3 , August 2016 , pp. 470 - 496
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Berman, A., Plemmons, R. J.: Nonnegative matrices in the mathematical sciences. Academic Press, New York, 1979.Google Scholar
[2]Bi, C. J., Ginting, V.: A residual-type a posteriori error estimate of finite volume element method for a quasi-linear elliptic problem. Numerische Mathematik 114(1), 107–132 (2009).Google Scholar
[3]Brenner, S. C., Scott, L. R.: The mathematical theory of finite element methods. Springer-Verlag, New York, 1994.Google Scholar
[4]Ciarlet, P. G.: The finite element method for elliptic problems (Studies in Mathematics and its Applications). North-Holland Publishing Co., Amsterdam, 4 (1978).Google Scholar
[5]Cao, W. X., Zhang, Z. M., Zou, Q. S.: Is 2k-conjecture valid for finite volume methods? SIAM J. Numer. Anal. 53, 942–962 (2015).Google Scholar
[6]Cao, W. X., Zhang, Z. M., Zou, Q. S.: Superconvergence of any order finite volume schemes for 1D general elliptic equations. J. Sci. Comput. 56, 566–590 (2013).Google Scholar
[7]Chen, Z., Wu, J., Xu, Y.: High-order finite volume methods for elliptic boundary value problems. Adv. Comput. Math. 37, 191–253 (2012).Google Scholar
[8]Chen, Z., Xu, Y., Zhang, Y.: Highter-order finite volume methods II: Inf-sup condition and uniform local ellipticity. J. Comput. and Applied. Math. 265, 96–109 (2014).CrossRefGoogle Scholar
[9]Chen, Z., Xu, Y., Zhang, Y.: A construction of highter-order finite volume methods. Math. Comp. 84(292), 599–628 (2014).Google Scholar
[10]Robert, Eymard, Thierry, Gallouët, Raphaèle, Hrebin: Finite volume methods (Handbook of Numerical Analysis). 7, 713–1018 (2000).Google Scholar
[11]Furihata, D.: Finite-dierence schemes for nonlinear wave equation that inherit energy conservation property. J. Comput. Appl. Math. 134, 37–57 (2001).Google Scholar
[12]Fu, S. W., Huang, S. K., Li, Y. S.: Numerical simulation of indirectly driven high-convergence implosions. Chinese J. Comput. Phys. 16, 162–166 (1999).Google Scholar
[13]Ham, F. E., Lien, F. S., Strong, A. B.: A fully conservative second-order finite difference scheme for incompressible flow on nonuniform grids. J. Comput. Phys. 177, 117–133 (2002).Google Scholar
[14]Harte, J. A., Alley, W. E., Bailey, D. S., Eddleman, J. L., Zimmerman, G. B.: Lasnex-a 2-d physics code for modeling icf. UCRL-LR-105821-96-4, 150–164 (1996).Google Scholar
[15]Jiang, J., Huang, Y. Q., Shu, S., Zeng, S.: Some new discretization and adaptation and multigrid methods for 2-D 3-T diffusion equations. J. Comput. Phys. 224, 168–181 (2007).Google Scholar
[16]Langer, S. H., Scott, H. A., Marinak, M. M., Landen, O. L.: Comparisons of line emission from 2- and 3- dimension simulations of icf capsules to experiments. J. Quant. Spectrosc. Radiat. Transfer, 275–286 (2003).Google Scholar
[17]Le Potier, C.: Finite volume monotone scheme for highly anisotropic diffusion operators on unstructured triangular meshes. C. R. Acad. Sci. Paris, Ser. I 341, 787–792 (2005).Google Scholar
[18]Lipnikov, K., Shashkov, M., Svyatskiy, D., Vassilevski, Yu.: Monotone finite volume schemes for diffusion equations on unstructured triangular and shape-regular polygonal meshes. J. Comput. Phys. 227, 492–512 (2007).Google Scholar
[19]Liska, R., Shashkov, M.: Enforcing the discrete maximum principle for linear finite element solutions of second-order elliptic problems. Commun. Comput. Phys. 3, 852–877 (2008).Google Scholar
[20]Li, R., Chen, Z., Wu, W.: The generalized difference methods for differential equations (Numerical analysis of finite volume methods). Marcel Dekker, New York, 2000.Google Scholar
[21]Li, R., Zhu, P.: Generalized difference methods for second order elliptic partial differential equations. I. The case of a triangular mesh. Numerical Mathematics. A Journal of Chinese Universities 4, 360–375 (1982).Google Scholar
[22]Lin, Y. P., Yang, M., Zou, Q. S.: L-2 error estimates for a class of any order finite volume schemes over quadrilateral meshes. SIAM J. Numer. Anal. 53, 2030–2050 (2015).Google Scholar
[23]Loubere, R. L., Staley, M., Wendroff, B.: The repair paradigm: New algorithms and applications to compressible flow. J. Comput. Phys. 211, 385–404 (2006).Google Scholar
[24]Lu, C. N., Huang, W. Z., Van Vleck, E. S.: The cutoff method for the numerical computation of nonnegative solutions of parabolic PDEs with application to anisotropic diffusion and lubrication-type equations. J. Comput. Phys. 242, 24–36 (2013).Google Scholar
[25]Mikaelian, K. O.: LASNEX simulations of the classical and laser-driven RayleighC-Taylor instability. Phys. Rev. 42, 4944–4951 (1990).Google Scholar
[26]Mo, Z. Y., Shen, L. J., Gabriel, W.: Parallel adaptive multigrid algorithm for 2-d 3-t diffusion equations. Int. J. Comput. Math. 81, 361–374 (2004).Google Scholar
[27]Nie, C. Y., Shu, S.: A kind of symmetric finite volume scheme for the two-dimension three-temperature heat conduction equations. Natural Science Journal of Xiangtan University 26, 18–22 (2004).Google Scholar
[28]Nie, C. Y., Tan, M.: Some discussion on how to solve the two-dimension three-temperature heat conduction equations. Comm. On Appl. Math. And Comput. 22(1), 57–61 (2008).Google Scholar
[29]Ober, C. C., Shadid, J. N.: Studies on the accuracy of time-integration methods for the radiation-diffusion equations. J. Comput. Phys. 195, 743–772 (2004).Google Scholar
[30]Perot, B.: Conservation properties of unstructured staggered mesh Schemes. J. Comput. Phys. 159, 58–89 (2002).Google Scholar
[31]Shashkov, M., Wendroff, B.: The repair paradigm and application to conservation laws. J. Comput. Phys. 198, 265–277 (2004).Google Scholar
[32]Sheng, Z. Q., Yuan, G. W.: A nine point scheme for the approximation of diffusion operators on distorted quadrilateral meshes. SIAM J. Sci. Comput. 30, 1341–1361 (2008).Google Scholar
[33]Stals, L.: Comparison of non-linear solvers for the solution of radiation transport equations. Electronic Transactions on Numerical Analysis 15, 78–93 (2003).Google Scholar
[34]Sheng, Z. Q., Yue, J. Y., Yuan, G. W.: Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes. SIAM J. Sci. Comput. 31, 2915–2934 (2009).Google Scholar
[35]Varga, R.: On a discrete maximum principle. SIAM J. Numer. Anal. 3, 355–359 (1966).Google Scholar
[36]Wang, S., Yuan, G. W., Li, Y. H., Sheng, Z. Q.: Discrete maximum principle based on repair technique for diamond type scheme of diffusion problems. Int. J. Numer. Meth. Fluids 70, 1188–1205 (2012).Google Scholar
[37]Wu, H. J., Li, Y. H., Li, R. H.: Adaptive generalized difference finite volume computations for two dimensional nonlinear parabolic equation. Chinese J. Comput. Phys. 20, 298–306 (2003).Google Scholar
[38]Xu, X. W., Mo, Z. Y., An, H. B.: Algebraic two-level iterative method for 2-D 3-T radiation diffusion equations. Chinese J. Comput. Phys. 26, 1–8 (2009).Google Scholar
[39]Xu, J., Zikatanov, L.: A monotone finite element scheme for convection diffusion equations. Math. Comp. 68, 1429–1446 (1999).CrossRefGoogle Scholar
[40]Xu, J. C., Zou, Q. S.: Analysis of linear and qualratic simplicial finite volume methods for elliptic equations. Numerische Mathematik 111, 469–492 (2009).Google Scholar
[41]Xu, J. C., Zhang, Z. M.: Analysis of recovery type a positeriori error estimators for mildly structured grids. Mathematics of Computation 73(247), 1139–1152 (2004).CrossRefGoogle Scholar
[42]Yang, X. B., Huang, W. Z., Qiu, J. X.: A moving mesh finite difference method for equilibrium radiation diffusion equations. J. Comput. Phys. 298, 661–677 (2015).Google Scholar
[43]Zhao, Q., Yuan, G. W.: Analysis and construction of cell-centered finite volume scheme for diffusion equations on distorted meshes. Comput. Meth. Appl. Mech. Engrg. 198, 3039–3050 (2009).Google Scholar
[44]Zhao, X. K., Chen, Y. L., Gao, Y. N., Yu, C. H., Li, Y. H.: Finite volume element methods for nonequilibrium radiation diffusion equations. Int. J. Numer. Meth. Fluids 73, 1059–1080 (2013).Google Scholar
[45]Zou, Q. S., Veeser, A., Kornhuber, R., Graser, C.: Hierarchical error estimates for the energy functional in obstacle problems. Numerische Methematik 117(4), 653–677 (2001).Google Scholar
[46]Zou, Q. S.: Hierarchical error estimates for finite volume approximation solution of elliptic equations. Applied Numerical Mathematics 60, 142–153 (2010).CrossRefGoogle Scholar
[47]Zou, Q. S., Kornhuber, R.: Efficient and reliable hierarchical error estimates for the discretization error of elliptic obstacle problems. Math. Comp. 80, 69–88 (2011).Google Scholar
[48]Zhang, Z. M., Yan, N. N.: Recovery type a posteriori error estimates in finite element methods. Korean J. Comput. & Appl. Math. 8(2), 235–251 (2001).Google Scholar
[49]Zhang, Z. M., Zou, Q. S.: A family of finite volume schemes of arbitrary order on rectangular meshes. J. Sci. Comput. 58, 308–330 (2014).Google Scholar
[50]Zhang, Z. M., Zou, Q. S.: Vertex-centered finite volume schemes of any order over quadrilateral meshes for elliptic boundary value problems. Numer. Math. 130, 363–393 (2015).Google Scholar