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Weighted Interior Penalty Method with Semi-Implicit Integration Factor Method for Non-Equilibrium Radiation Diffusion Equation

Published online by Cambridge University Press:  03 June 2015

Rongpei Zhang ,
Xijun Yu ,
Jiang Zhu ,
Abimael F. D. Loula  and
Xia Cui
Rongpei Zhang*
Affiliation:
School of Sciences, Liaoning ShiHua University, Fushun 113001, China Laboratório Nacional de Computação Científica, MCTI, Avenida Getúlio Vargas 333, 25651-075 Petrópolis, RJ, Brazil
Xijun Yu*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
Jiang Zhu*
Affiliation:
Laboratório Nacional de Computação Científica, MCTI, Avenida Getúlio Vargas 333, 25651-075 Petrópolis, RJ, Brazil
Abimael F. D. Loula*
Affiliation:
Laboratório Nacional de Computação Científica, MCTI, Avenida Getúlio Vargas 333, 25651-075 Petrópolis, RJ, Brazil
Xia Cui*
Affiliation:
Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
*
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
Email:[email protected]
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Abstract

Weighted interior penalty discontinuous Galerkin method is developed to solve the two-dimensional non-equilibrium radiation diffusion equation on unstructured mesh. There are three weights including the arithmetic, the harmonic, and the geometric weight in the weighted discontinuous Galerkin scheme. For the time discretization, we treat the nonlinear diffusion coefficients explicitly, and apply the semi-implicit integration factor method to the nonlinear ordinary differential equations arising from discontinuous Galerkin spatial discretization. The semi-implicit integration factor method can not only avoid severe timestep limits, but also takes advantage of the local property of DG methods by which small sized nonlinear algebraic systems are solved element by element with the exact Newton iteration method. Numerical results are presented to demonstrate the validity of discontinuous Galerkin method for high nonlinear and tightly coupled radiation diffusion equation.

Keywords

35R0535Q7965M6080A20Discontinuous Galerkinweighted interior penaltysemi-implicit integration factornon-equilibrium radiation diffusion
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 5 , November 2013 , pp. 1287 - 1303
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Bowers, R.L., Wilson, J.R., Numerical Modeling in Applied Physics and Astrophysics, Jones and Bartlett, Boston Publishers, 1991.Google Scholar
[2]Robinson, A.C., Garasi, C.J., Three-dimensional z-pinch wire array modeling with ALEGRA-HEDP, Comput. Phys. Commun. 164 (2004) 408–413.CrossRefGoogle Scholar
[3]Turner, N.J., Stone, J.M., A module for radiation hydrodynamic calculations with ZEUS-2D using flux-limited diffusion, Astrphys. J. Suppl. Ser. 135 (2001) 95–107.CrossRefGoogle Scholar
[4]Knoll, D.A., Rider, W.J., Olson, G.L., Aneffcient nonlinear solution method for nonequilibrium radiation diffusion, J. Quantum Spectrosc. Radiat. Transfer 63 (1999) 15–29.CrossRefGoogle Scholar
[5]Knoll, D.A., Rider, W.J., and Olson, G.L., Nonlinear convergence, accuracy, and time step control in nonequilibrium radiation diffusion, J. Quant. Spectrosc. Radiat. Transfer 70 (2001) 25–36.CrossRefGoogle Scholar
[6]Mousseau, V.A., Knoll, D.A. and Rider, W.J., Physics-based preconditioning and the Newton-Krylov method for non-linear-equilibrium radiation diffusion, J. Comput. Phys. 160 (2000) 743–765.CrossRefGoogle Scholar
[7]Mousseau, V.A., Knoll, D.A., New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion, J. Comput. Phys. 190 (2003) 42–51.Google Scholar
[8]Kadioglu, S.Y., Knoll, D.A., Lowrie, R.B., Rauenzahn, R.M., A second order self-consistent IMEX method for radiation hydrodynamics, J. Comput. Phys. 229 (2010) 8313–8332.Google Scholar
[9]Sheng, Z.Q., Yue, J.Y. and Yuan, G.W., Monotone finite volume schemes of nonequilibrium radiation diffusion equations on distorted meshes, SIAM J. Sci. Comput. 31 (2009) 2915–2934.CrossRefGoogle Scholar
[10]Yuan, G. W., Hang, X. D., Sheng, Z. Q., Yue, J. Y.Progress in numerical methods for radiation diffusion equations. Chinese J. Comput. Phys. 26 (2009) 475–500.Google Scholar
[11]Yue, J.Y., Yuan, G.W., Picard-Newton iterative method with time step control for multimaterial non-equilibrium radiation diffusion problem, Commun. Comput. Phys. 10 (2011) 844–866.CrossRefGoogle Scholar
[12]Mavriplis, D.J., Multigrid approaches to non-linear diffusion problems on unstructured meshes, Numer. Linear Algebra Appl. 8 (2001) 499–512.Google Scholar
[13]Kang, K.E., P1 nonconforming finite element multigrid method for radiation transport, SIAM J Sci Comput, 25 (2003) 369–384.CrossRefGoogle Scholar
[14]Reed, W.H., Hill, T.R., Triangular mesh methods for the neutron transport equation, Los Alamos Scienfic Laboratory Report, LA2UR2732479, 1973.Google Scholar
[15]Cockburn, B., Shu, C. W.The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998) 2440–2463.Google Scholar
[16]Gassner, G., F. Lörcher, and Munz, C. A.A contribution to the construction of diffusion fluxes for finite volume and discontinuous Galerkin schemes. J. Comput. Phys. 224 (2007) 1049–1063.CrossRefGoogle Scholar
[17]F. Lörcher, , Gassner, G. and Munz, C. A.An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comput. Phys. 227 (2008) 5649–5670.CrossRefGoogle Scholar
[18]Liu, H., Yan, J.The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47 (2009), 675–698.CrossRefGoogle Scholar
[19]Liu, H., Yan, J.The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 3 (2010) 541–564.Google Scholar
[20]Song, L.Fully discrete interior penalty discontinuous Galerkin methods for nonlinear parabolic equations, Numer. Method Partial. Differ. E. 2010 DOI:10.1002/num.20619.Google Scholar
[21]Ern, A., Stephansen, A. and Zunino, P.A discontinuous Galerkin method with weighted averages for advection-diffusion equations with locally small and anisotropic diffusivity. IMA J. Numer. Ana. 29(2) (2009) 235–256.Google Scholar
[22]Cai, Z., Ye, X. and Zhang, S, Discontinuous Galerkin Finite Element Methods for Interface Problems: A Priori and A Posteriori Error Estimations. SIAM J. Numer. Anal. 49 (2011) 1761–1787.CrossRefGoogle Scholar
[23]Lowrie, R.B., A comparison of implicit time integration method for nonlinear relaxation and diffusion, J. Comput. Phys. 196 (2004) 566–590.Google Scholar
[24]Knoll, D.A., Lowrie, R.B., Morel, J.E., Numerical analysis of time integration errors for nonequilibrium radiation diffusion. J. Comput. Phys. 226 (2007) 1332–1347.CrossRefGoogle Scholar
[25]Mousseau, V.A., Knoll, D.A., Temporal accuracy of the nonequilibrium radiation diffusion equations applied to two-dimensional multimaterial simulations, Nuclear Science and Engineering. 154 (2006) 174–189.Google Scholar
[26]Verwer, J.G., Sommeijer, B.P., An Implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations, SIAM J. Sci. Comput. 25 (2004) 1824–1835.Google Scholar
[27]Nie, Q., Zhang, Y.-T., Zhao, R., Efficient semi-implicit schemes for stiff systems, J. Comput. Phy. 214 (2006) 521–537.Google Scholar
[28]Chen, S., Zhang, Y., Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods, J. Comput. Phy. 230 (2011) 4336–4352.CrossRefGoogle Scholar
[29]Sidje, R. B., Expokit: Software package for computing matrix exponentials, ACM Trans. Math. Software 24 (1998) 130–156.CrossRefGoogle Scholar
[30]Yuan, G. W., Sheng, Z. Q.Analysis of accuracy of a finite volume scheme for diffusion equations on distorted meshes, J. Comput. Phys. 224 (2007) 1170–1189.Google Scholar