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doi: 10.3934/krm.2022014
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A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis

CMAP, École Polytechnique, CNRS UMR7641, Institut Polytechnique de Paris, Route de Saclay, Palaiseau, 91128, France

Received  January 2022 Revised  April 2022 Early access May 2022

Fund Project: The author is supported by DGA AID through MMEED project

A closure relation for moments equations in kinetic theory was recently introduced in [38], based on the study of the geometry of the set of moments. This relation was constructed from a projection of a moment vector toward the boundary of the set of moments and corresponds to approximating the underlying kinetic distribution as a sum of a chosen equilibrium distribution plus a sum of purely anisotropic Dirac distributions.

The present work generalizes this construction for kinetic equations involving unbounded velocities, i.e. to the Hamburger problem, and provides a deeper analysis of the resulting moment system. Especially, we provide representation results for moment vectors along the boundary of the moment set that implies the well-definition of the model. And the resulting moment model is shown to be weakly hyperbolic with peculiar properties of hyperbolicity and entropy of two subsystems, corresponding respectively to the equilibrium and to the purely anisotropic parts of the underlying kinetic distribution.

Citation: Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis. Kinetic and Related Models, doi: 10.3934/krm.2022014
References:
[1]

M. Abdel Malik, Adaptive Algorithms for Optimal Multiscale Model Hierarchies of the Boltzmann Equation: Galerkin Methods for Kinetic Theory, PhD thesis, Mechanical Engineering, May 2017.

[2]

M. Abdel Malik and H. van Brummelen, Moment closure approximations of the boltzmann equation based on φ-divergences, J. Stat. Phys., 164 (2016), 77-104.  doi: 10.1007/s10955-016-1529-5.

[3]

N. I. Akhiezer, The Classical Moment Problem, Edinburgh : Oliver & Boyd, 1965.

[4]

G. W. Alldredge, Optimization Techniques for Entropy-Based Moment Models of LinearTransport, PhD thesis, University of Maryland, 2012.

[5]

G. W. AlldredgeC. 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 J. Sci. Comput., 34 (2012), 361-391.  doi: 10.1137/11084772X.

[6]

J. Borwein and A. Lewis, Duality relationships for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), 325-338.  doi: 10.1137/0329017.

[7]

F. Bouchut, On zero pressure gas dynamics, Advances in Kin. Theory and Comput., 22 (1994), 171-190. 

[8]

Z. CaiY. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518.  doi: 10.1002/cpa.21472.

[9]

C. Chalons and D. Kah M. Massot, Beyond pressureless gas dynamics : Quadrature-based velocity moment models, Commun. Math. Sci., 10 (2012), 1241-1272.  doi: 10.4310/CMS.2012.v10.n4.a11.

[10]

G.-Q. ChenC. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., 47 (1994), 787-830.  doi: 10.1002/cpa.3160470602.

[11]

R. Curto and L. Fialkow, Recursiveness, positivity, and truncated moment problem, Houston J. Math., 17 (1991), 603-635. 

[12]

R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Memoirs Amer. Math. Soc., 136 (1998), 1-56.  doi: 10.1090/memo/0648.

[13]

R. Curto and L. Fialkow, A duality proof of Tchakaloff's theorem, J. Math. Anal. Appl., 269 (2002), 519-532.  doi: 10.1016/S0022-247X(02)00034-3.

[14]

B. Dubroca and J.-L. Feugeas, Hiérarchie des modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.

[15]

L. Fialkow, The truncated K-moment problem: A survey, Theta Ser. Adv. Math., 18 (2016), 25-51. 

[16]

R. Fox and F. Laurent, Hyperbolic quadrature method of moments for the one-dimensional kinetic equation, SIAM J. Appl. Math., 82 (2022), 750-771.  doi: 10.1137/21M1406143.

[17]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci., 68 (1971), 1686-1688.  doi: 10.1073/pnas.68.8.1686.

[18]

S. K. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523. 

[19]

H. Grad, On the kinetic theory of rarefied gases, Commun. Pure and Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[20]

H. Hamburger, Über eine Erweiterung des Stieltjesschen Momentenproblems, Math. Ann., 82 (1920), 120-164.  doi: 10.1007/BF01457982.

[21]

C. HauckC. Levermore and A. Tits, Convex duality and entropy-based moment closures: Characterizing degenerate densities, SIAM J. Control Optim, 47 (2008), 1977-2015.  doi: 10.1137/070691139.

[22]

C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.  doi: 10.4310/CMS.2011.v9.n1.a9.

[23]

F. Hausdorff, Summationmethoden und Momentfolgen. I., Math. Z., 9 (1921), 74-109.  doi: 10.1007/BF01378337.

[24]

M. Junk, Domain of definition of Levermore's five-moment system, J. Stat. Phys., 93 (1998), 1143-1167.  doi: 10.1023/B:JOSS.0000033155.07331.d9.

[25]

M. Junk, Maximum entropy for reduced moment problems, Math. Models Methods Appl. Sci., 10 (2000), 1001-1025.  doi: 10.1142/S0218202500000513.

[26]

S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 174 (2004), 345-364.  doi: 10.1007/s00205-004-0330-9.

[27] J.-B. Lasserre, Moment, Positive Polynomials, and Their Applications, Imperial College Press, London, 2010. 
[28]

P. G. Le Floch, An existence and uniqueness result for two nonstrictly hyperbolic systems, Nonlinear Evolution Equations That Change Type, 27 (1990), 126-138.  doi: 10.1007/978-1-4613-9049-7_10.

[29] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.
[30]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[31]

D. Marchisio and R. Fox, Solution of population balance equations using the direct quadrature method of moments, J. Aerosol Sci., 36 (2005), 43-73.  doi: 10.1016/j.jaerosci.2004.07.009.

[32] D. Marchisio and R. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139016599.
[33]

J. McDonald and M. Torrilhon, Affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.

[34]

R. McGraw, Description of aerosol dynamics by the quadrature method of moments, Aerosol S. and Tech., 27 (1997), 255-265.  doi: 10.1080/02786829708965471.

[35]

L. R. Mead and N. Papanicolaou, Maximum entropy in the problem of moments, J. Math. Phys., 25 (1984), 2404-2417.  doi: 10.1063/1.526446.

[36]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectros. Radiat. Transfer, 20 (1978), 541-545.  doi: 10.1016/0022-4073(78)90024-9.

[37]

M. S. Mock, Systems of conservation laws of mixed type, J. Diff. Eq., 37 (1980), 70-88.  doi: 10.1016/0022-0396(80)90089-3.

[38]

T. Pichard, A moment closure based on a projection on the boundary of the realizability domain: 1d case, Kinet. Relat. Models, 13 (2020), 1243-1280.  doi: 10.3934/krm.2020045.

[39]

T. PichardG. W. AlldredgeS. BrullB. Dubroca and M. Frank, An approximation of the M2 closure: Application to radiotherapy dose simulation, J. Sci. Comput., 71 (2017), 71-108.  doi: 10.1007/s10915-016-0292-8.

[40]

A. ViéR. Fox and F. Laurent, Conditional hyperbolic quadrature method of moments for kinetic equations, J. Comput. Phys., 365 (2018), 269-293.  doi: 10.1016/j.jcp.2018.03.025.

[41]

M. Riesz, Sur le problème des moments, troisiéme note, Ark. Math. Astr. Fys., 17 (1923), 1-52. 

[42]

J. Schneider, Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561.  doi: 10.1051/m2an:2004025.

[43]

T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), 1-122.  doi: 10.5802/afst.108.

[44]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680.  doi: 10.1063/1.1597472.

[45]

W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Rational Mech. Anal., 172 (2004), 247-266.  doi: 10.1007/s00205-003-0304-3.

[46]

C. Yuan and R. Fox, Conditional quadrature method of moments for kinetic equations, J. Comput. Phys., 230 (2011), 8216-8246.  doi: 10.1016/j.jcp.2011.07.020.

[47]

C. YuanF. Laurent and R. O. Fox, An extended quadrature method of moments for population balance equations, J. Aerosol Sci., 51 (2012), 1-23. 

show all references

References:
[1]

M. Abdel Malik, Adaptive Algorithms for Optimal Multiscale Model Hierarchies of the Boltzmann Equation: Galerkin Methods for Kinetic Theory, PhD thesis, Mechanical Engineering, May 2017.

[2]

M. Abdel Malik and H. van Brummelen, Moment closure approximations of the boltzmann equation based on φ-divergences, J. Stat. Phys., 164 (2016), 77-104.  doi: 10.1007/s10955-016-1529-5.

[3]

N. I. Akhiezer, The Classical Moment Problem, Edinburgh : Oliver & Boyd, 1965.

[4]

G. W. Alldredge, Optimization Techniques for Entropy-Based Moment Models of LinearTransport, PhD thesis, University of Maryland, 2012.

[5]

G. W. AlldredgeC. 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 J. Sci. Comput., 34 (2012), 361-391.  doi: 10.1137/11084772X.

[6]

J. Borwein and A. Lewis, Duality relationships for entropy-like minimization problems, SIAM J. Control Optim., 29 (1991), 325-338.  doi: 10.1137/0329017.

[7]

F. Bouchut, On zero pressure gas dynamics, Advances in Kin. Theory and Comput., 22 (1994), 171-190. 

[8]

Z. CaiY. Fan and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), 464-518.  doi: 10.1002/cpa.21472.

[9]

C. Chalons and D. Kah M. Massot, Beyond pressureless gas dynamics : Quadrature-based velocity moment models, Commun. Math. Sci., 10 (2012), 1241-1272.  doi: 10.4310/CMS.2012.v10.n4.a11.

[10]

G.-Q. ChenC. D. Levermore and T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., 47 (1994), 787-830.  doi: 10.1002/cpa.3160470602.

[11]

R. Curto and L. Fialkow, Recursiveness, positivity, and truncated moment problem, Houston J. Math., 17 (1991), 603-635. 

[12]

R. Curto and L. Fialkow, Flat extensions of positive moment matrices: Recursively generated relations, Memoirs Amer. Math. Soc., 136 (1998), 1-56.  doi: 10.1090/memo/0648.

[13]

R. Curto and L. Fialkow, A duality proof of Tchakaloff's theorem, J. Math. Anal. Appl., 269 (2002), 519-532.  doi: 10.1016/S0022-247X(02)00034-3.

[14]

B. Dubroca and J.-L. Feugeas, Hiérarchie des modèles aux moments pour le transfert radiatif, C. R. Acad. Sci. Paris Sér. I Math., 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.

[15]

L. Fialkow, The truncated K-moment problem: A survey, Theta Ser. Adv. Math., 18 (2016), 25-51. 

[16]

R. Fox and F. Laurent, Hyperbolic quadrature method of moments for the one-dimensional kinetic equation, SIAM J. Appl. Math., 82 (2022), 750-771.  doi: 10.1137/21M1406143.

[17]

K. O. Friedrichs and P. D. Lax, Systems of conservation equations with a convex extension, Proc. Nat. Acad. Sci., 68 (1971), 1686-1688.  doi: 10.1073/pnas.68.8.1686.

[18]

S. K. Godunov, An interesting class of quasilinear systems, Dokl. Akad. Nauk SSSR, 139 (1961), 521-523. 

[19]

H. Grad, On the kinetic theory of rarefied gases, Commun. Pure and Appl. Math., 2 (1949), 331-407.  doi: 10.1002/cpa.3160020403.

[20]

H. Hamburger, Über eine Erweiterung des Stieltjesschen Momentenproblems, Math. Ann., 82 (1920), 120-164.  doi: 10.1007/BF01457982.

[21]

C. HauckC. Levermore and A. Tits, Convex duality and entropy-based moment closures: Characterizing degenerate densities, SIAM J. Control Optim, 47 (2008), 1977-2015.  doi: 10.1137/070691139.

[22]

C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 9 (2011), 187-205.  doi: 10.4310/CMS.2011.v9.n1.a9.

[23]

F. Hausdorff, Summationmethoden und Momentfolgen. I., Math. Z., 9 (1921), 74-109.  doi: 10.1007/BF01378337.

[24]

M. Junk, Domain of definition of Levermore's five-moment system, J. Stat. Phys., 93 (1998), 1143-1167.  doi: 10.1023/B:JOSS.0000033155.07331.d9.

[25]

M. Junk, Maximum entropy for reduced moment problems, Math. Models Methods Appl. Sci., 10 (2000), 1001-1025.  doi: 10.1142/S0218202500000513.

[26]

S. Kawashima and W.-A. Yong, Dissipative structure and entropy for hyperbolic systems of balance laws, Arch. Rational Mech. Anal., 174 (2004), 345-364.  doi: 10.1007/s00205-004-0330-9.

[27] J.-B. Lasserre, Moment, Positive Polynomials, and Their Applications, Imperial College Press, London, 2010. 
[28]

P. G. Le Floch, An existence and uniqueness result for two nonstrictly hyperbolic systems, Nonlinear Evolution Equations That Change Type, 27 (1990), 126-138.  doi: 10.1007/978-1-4613-9049-7_10.

[29] R. J. LeVeque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253.
[30]

C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.

[31]

D. Marchisio and R. Fox, Solution of population balance equations using the direct quadrature method of moments, J. Aerosol Sci., 36 (2005), 43-73.  doi: 10.1016/j.jaerosci.2004.07.009.

[32] D. Marchisio and R. Fox, Computational Models for Polydisperse Particulate and Multiphase Systems, Cambridge University Press, 2013.  doi: 10.1017/CBO9781139016599.
[33]

J. McDonald and M. Torrilhon, Affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), 500-523.  doi: 10.1016/j.jcp.2013.05.046.

[34]

R. McGraw, Description of aerosol dynamics by the quadrature method of moments, Aerosol S. and Tech., 27 (1997), 255-265.  doi: 10.1080/02786829708965471.

[35]

L. R. Mead and N. Papanicolaou, Maximum entropy in the problem of moments, J. Math. Phys., 25 (1984), 2404-2417.  doi: 10.1063/1.526446.

[36]

G. N. Minerbo, Maximum entropy Eddington factors, J. Quant. Spectros. Radiat. Transfer, 20 (1978), 541-545.  doi: 10.1016/0022-4073(78)90024-9.

[37]

M. S. Mock, Systems of conservation laws of mixed type, J. Diff. Eq., 37 (1980), 70-88.  doi: 10.1016/0022-0396(80)90089-3.

[38]

T. Pichard, A moment closure based on a projection on the boundary of the realizability domain: 1d case, Kinet. Relat. Models, 13 (2020), 1243-1280.  doi: 10.3934/krm.2020045.

[39]

T. PichardG. W. AlldredgeS. BrullB. Dubroca and M. Frank, An approximation of the M2 closure: Application to radiotherapy dose simulation, J. Sci. Comput., 71 (2017), 71-108.  doi: 10.1007/s10915-016-0292-8.

[40]

A. ViéR. Fox and F. Laurent, Conditional hyperbolic quadrature method of moments for kinetic equations, J. Comput. Phys., 365 (2018), 269-293.  doi: 10.1016/j.jcp.2018.03.025.

[41]

M. Riesz, Sur le problème des moments, troisiéme note, Ark. Math. Astr. Fys., 17 (1923), 1-52. 

[42]

J. Schneider, Entropic approximation in kinetic theory, M2AN Math. Model. Numer. Anal., 38 (2004), 541-561.  doi: 10.1051/m2an:2004025.

[43]

T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 8 (1894), 1-122.  doi: 10.5802/afst.108.

[44]

H. Struchtrup and M. Torrilhon, Regularization of Grad's 13 moment equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), 2668-2680.  doi: 10.1063/1.1597472.

[45]

W.-A. Yong, Entropy and global existence for hyperbolic balance laws, Arch. Rational Mech. Anal., 172 (2004), 247-266.  doi: 10.1007/s00205-003-0304-3.

[46]

C. Yuan and R. Fox, Conditional quadrature method of moments for kinetic equations, J. Comput. Phys., 230 (2011), 8216-8246.  doi: 10.1016/j.jcp.2011.07.020.

[47]

C. YuanF. Laurent and R. O. Fox, An extended quadrature method of moments for population balance equations, J. Aerosol Sci., 51 (2012), 1-23. 

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