Skip to main content Accessibility help

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

  • Rongpei Zhang (a1) (a2), Xijun Yu (a3), Jiang Zhu (a2), Abimael F. D. Loula (a2) and Xia Cui (a3)...


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.


Corresponding author



Hide All
[1]Bowers, R.L., Wilson, J.R., Numerical Modeling in Applied Physics and Astrophysics, Jones and Bartlett, Boston Publishers, 1991.
[2]Robinson, A.C., Garasi, C.J., Three-dimensional z-pinch wire array modeling with ALEGRA-HEDP, Comput. Phys. Commun. 164 (2004) 408413.
[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) 95107.
[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) 1529.
[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) 2536.
[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) 743765.
[7]Mousseau, V.A., Knoll, D.A., New physics-based preconditioning of implicit methods for non-equilibrium radiation diffusion, J. Comput. Phys. 190 (2003) 4251.
[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) 83138332.
[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) 29152934.
[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) 475500.
[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) 844866.
[12]Mavriplis, D.J., Multigrid approaches to non-linear diffusion problems on unstructured meshes, Numer. Linear Algebra Appl. 8 (2001) 499512.
[13]Kang, K.E., P1 nonconforming finite element multigrid method for radiation transport, SIAM J Sci Comput, 25 (2003) 369384.
[14]Reed, W.H., Hill, T.R., Triangular mesh methods for the neutron transport equation, Los Alamos Scienfic Laboratory Report, LA2UR2732479, 1973.
[15]Cockburn, B., Shu, C. W.The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal. 35 (1998) 24402463.
[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) 10491063.
[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) 56495670.
[18]Liu, H., Yan, J.The direct discontinuous Galerkin (DDG) methods for diffusion problems. SIAM J. Numer. Anal. 47 (2009), 675698.
[19]Liu, H., Yan, J.The direct discontinuous Galerkin (DDG) method for diffusion with interface corrections. Commun. Comput. Phys. 3 (2010) 541564.
[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.
[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) 235256.
[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) 17611787.
[23]Lowrie, R.B., A comparison of implicit time integration method for nonlinear relaxation and diffusion, J. Comput. Phys. 196 (2004) 566590.
[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) 13321347.
[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) 174189.
[26]Verwer, J.G., Sommeijer, B.P., An Implicit-explicit Runge-Kutta-Chebyshev scheme for diffusion-reaction equations, SIAM J. Sci. Comput. 25 (2004) 18241835.
[27]Nie, Q., Zhang, Y.-T., Zhao, R., Efficient semi-implicit schemes for stiff systems, J. Comput. Phy. 214 (2006) 521537.
[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) 43364352.
[29]Sidje, R. B., Expokit: Software package for computing matrix exponentials, ACM Trans. Math. Software 24 (1998) 130156.
[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) 11701189.



Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed