June  2016, 9(2): 237-249. doi: 10.3934/krm.2016.9.237

Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry

1. 

Department of Mathematics RWTH Aachen University, Aachen, Germany

2. 

CAPT, LMAM & School of Mathematical Sciences, Peking University, Beijing, China

3. 

School of Mathematical Sciences, Peking University, Beijing, China

Received  January 2015 Revised  October 2015 Published  March 2016

We consider the simplest member of the hierarchy of the extended quadrature method of moments (EQMOM), which gives equations for the zeroth-, first-, and second-order moments of the energy density of photons in the radiative transfer equations in slab geometry. First we show that the equations are well-defined for all moment vectors consistent with a nonnegative underlying distribution, and that the reconstruction is explicit and therefore computationally inexpensive. Second, we show that the resulting moment equations are hyperbolic. These two properties make this moment method quite similar to the attractive but far more expensive $M_2$ method. We confirm through numerical solutions to several benchmark problems that the methods give qualitatively similar results.
Citation: Graham W. Alldredge, Ruo Li, Weiming Li. Approximating the $M_2$ method by the extended quadrature method of moments for radiative transfer in slab geometry. Kinetic & Related Models, 2016, 9 (2) : 237-249. doi: 10.3934/krm.2016.9.237
References:
[1]

G. W. Alldredge, C. D. Hauck, D. P. O'Leary and A. L. Tits, Adaptive change of basis in entropy-based moment closures for linear kinetic equations,, Journal of Computational Physics, 258 (2014), 489.  doi: 10.1016/j.jcp.2013.10.049.  Google Scholar

[2]

G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry ii: A computational study of the optimization problem,, SIAM Journal on Scientific Computing, 34 (2012).  doi: 10.1137/11084772X.  Google Scholar

[3]

C. Berthon, P. Charrier and B. Dubroca, An hllc scheme to solve the $M_1$ model of radiative transfer in two space dimensions,, Journal of Scientific Computing, 31 (2007), 347.  doi: 10.1007/s10915-006-9108-6.  Google Scholar

[4]

T. A. Brunner, Forms of Approximate Radiation Transport,, Tech. Rep SAND2002-1778, (2002), 2002.   Google Scholar

[5]

T. A. Brunner and J. P. Holloway, One-dimensional riemann solvers and the maximum entropy closure,, Journal of Quantitative Spectroscopy and Radiative Transfer, 69 (2001), 543.  doi: 10.1016/S0022-4073(00)00099-6.  Google Scholar

[6]

R. Curto and L. Fialkow, Recursiveness, positivity and truncated moment problems,, Houston J. Math, 17 (1991), 603.   Google Scholar

[7]

B. Dubroca and J.-L. Fuegas, Étude théorique et numérique d'une hiérarchie de modèles aus moments pour le transfert radiatif,, C.R. Acad. Sci. Paris, 329 (1999), 915.  doi: 10.1016/S0764-4442(00)87499-6.  Google Scholar

[8]

C. K. Garrett, C. Hauck and J. Hill, Optimization and large scale computation of an entropy-based moment closure,, Journal of Computational Physics, 302 (2015), 573.   Google Scholar

[9]

C. D. Levermore, Moment closure hierarchies for kinetic theories,, Journal of Statistical Physics, 83 (1996), 1021.  doi: 10.1007/BF02179552.  Google Scholar

[10]

E. E. Lewis and W. F. Miller, Jr, Computational Methods in Neutron Transport,, John Wiley and Sons, (1984).   Google Scholar

[11]

R. G. McClarren, J. P. Holloway and T. A. Brunner, On solutions to the $P_n$ equations for thermal radiative transfer,, Journal of Computational Physics, 227 (2008), 2864.  doi: 10.1016/j.jcp.2007.11.027.  Google Scholar

[12]

E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous galerkin method for the m1 model of radiative transfer,, Journal of Computational Physics, 231 (2012), 5612.  doi: 10.1016/j.jcp.2012.03.002.  Google Scholar

[13]

G. C. Pomraning, The Equations of Radiation Hydrodynamics,, Courier Dover Publications, (1973).   Google Scholar

[14]

B. Su and G. L. Olson, An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,, Annals of Nuclear Energy, 24 (1997), 1035.  doi: 10.1016/S0306-4549(96)00100-4.  Google Scholar

[15]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction,, Springer Science & Business Media, (2009).  doi: 10.1007/b79761.  Google Scholar

[16]

V. Vikas, C. D. Hauck, Z. J. Wang and R. O. Fox, Radiation transport modeling using extended quadrature method of moments,, Journal of Computational Physics, 246 (2013), 221.  doi: 10.1016/j.jcp.2013.03.028.  Google Scholar

show all references

References:
[1]

G. W. Alldredge, C. D. Hauck, D. P. O'Leary and A. L. Tits, Adaptive change of basis in entropy-based moment closures for linear kinetic equations,, Journal of Computational Physics, 258 (2014), 489.  doi: 10.1016/j.jcp.2013.10.049.  Google Scholar

[2]

G. W. Alldredge, C. D. Hauck and A. L. Tits, High-order entropy-based closures for linear transport in slab geometry ii: A computational study of the optimization problem,, SIAM Journal on Scientific Computing, 34 (2012).  doi: 10.1137/11084772X.  Google Scholar

[3]

C. Berthon, P. Charrier and B. Dubroca, An hllc scheme to solve the $M_1$ model of radiative transfer in two space dimensions,, Journal of Scientific Computing, 31 (2007), 347.  doi: 10.1007/s10915-006-9108-6.  Google Scholar

[4]

T. A. Brunner, Forms of Approximate Radiation Transport,, Tech. Rep SAND2002-1778, (2002), 2002.   Google Scholar

[5]

T. A. Brunner and J. P. Holloway, One-dimensional riemann solvers and the maximum entropy closure,, Journal of Quantitative Spectroscopy and Radiative Transfer, 69 (2001), 543.  doi: 10.1016/S0022-4073(00)00099-6.  Google Scholar

[6]

R. Curto and L. Fialkow, Recursiveness, positivity and truncated moment problems,, Houston J. Math, 17 (1991), 603.   Google Scholar

[7]

B. Dubroca and J.-L. Fuegas, Étude théorique et numérique d'une hiérarchie de modèles aus moments pour le transfert radiatif,, C.R. Acad. Sci. Paris, 329 (1999), 915.  doi: 10.1016/S0764-4442(00)87499-6.  Google Scholar

[8]

C. K. Garrett, C. Hauck and J. Hill, Optimization and large scale computation of an entropy-based moment closure,, Journal of Computational Physics, 302 (2015), 573.   Google Scholar

[9]

C. D. Levermore, Moment closure hierarchies for kinetic theories,, Journal of Statistical Physics, 83 (1996), 1021.  doi: 10.1007/BF02179552.  Google Scholar

[10]

E. E. Lewis and W. F. Miller, Jr, Computational Methods in Neutron Transport,, John Wiley and Sons, (1984).   Google Scholar

[11]

R. G. McClarren, J. P. Holloway and T. A. Brunner, On solutions to the $P_n$ equations for thermal radiative transfer,, Journal of Computational Physics, 227 (2008), 2864.  doi: 10.1016/j.jcp.2007.11.027.  Google Scholar

[12]

E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous galerkin method for the m1 model of radiative transfer,, Journal of Computational Physics, 231 (2012), 5612.  doi: 10.1016/j.jcp.2012.03.002.  Google Scholar

[13]

G. C. Pomraning, The Equations of Radiation Hydrodynamics,, Courier Dover Publications, (1973).   Google Scholar

[14]

B. Su and G. L. Olson, An analytical benchmark for non-equilibrium radiative transfer in an isotropically scattering medium,, Annals of Nuclear Energy, 24 (1997), 1035.  doi: 10.1016/S0306-4549(96)00100-4.  Google Scholar

[15]

E. F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction,, Springer Science & Business Media, (2009).  doi: 10.1007/b79761.  Google Scholar

[16]

V. Vikas, C. D. Hauck, Z. J. Wang and R. O. Fox, Radiation transport modeling using extended quadrature method of moments,, Journal of Computational Physics, 246 (2013), 221.  doi: 10.1016/j.jcp.2013.03.028.  Google Scholar

[1]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[2]

Timothy Chumley, Renato Feres. Entropy production in random billiards. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1319-1346. doi: 10.3934/dcds.2020319

[3]

Bing Gao, Rui Gao. On fair entropy of the tent family. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021017

[4]

Giuseppe Capobianco, Tom Winandy, Simon R. Eugster. The principle of virtual work and Hamilton's principle on Galilean manifolds. Journal of Geometric Mechanics, 2021  doi: 10.3934/jgm.2021002

[5]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[6]

San Ling, Buket Özkaya. New bounds on the minimum distance of cyclic codes. Advances in Mathematics of Communications, 2021, 15 (1) : 1-8. doi: 10.3934/amc.2020038

[7]

Tahir Aliyev Azeroğlu, Bülent Nafi Örnek, Timur Düzenli. Some results on the behaviour of transfer functions at the right half plane. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020106

[8]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

[9]

Xinlin Cao, Huaian Diao, Jinhong Li. Some recent progress on inverse scattering problems within general polyhedral geometry. Electronic Research Archive, 2021, 29 (1) : 1753-1782. doi: 10.3934/era.2020090

[10]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325

[11]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048

[12]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[13]

Mark F. Demers. Uniqueness and exponential mixing for the measure of maximal entropy for piecewise hyperbolic maps. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 217-256. doi: 10.3934/dcds.2020217

[14]

Antoine Benoit. Weak well-posedness of hyperbolic boundary value problems in a strip: when instabilities do not reflect the geometry. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5475-5486. doi: 10.3934/cpaa.2020248

[15]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[16]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[17]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266

[18]

Magdalena Foryś-Krawiec, Jiří Kupka, Piotr Oprocha, Xueting Tian. On entropy of $ \Phi $-irregular and $ \Phi $-level sets in maps with the shadowing property. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1271-1296. doi: 10.3934/dcds.2020317

[19]

Simone Göttlich, Elisa Iacomini, Thomas Jung. Properties of the LWR model with time delay. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020032

[20]

Ténan Yeo. Stochastic and deterministic SIS patch model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021012

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (70)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]