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December  2017, 10(4): 1127-1161. doi: 10.3934/krm.2017044

First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions

Florian Schneider , , Andreas Roth and  Jochen Kall

Technische Universität Kaiserslautern, 67663 Kaiserslautern, Germany

* Corresponding author: Florian Schneider

Received  August 2016 Revised  November 2016 Published  March 2017

Mixed-moment models, introduced in [8,44] for one space dimension, are a modification of the method of moments applied to a (linear) kinetic equation, by choosing mixtures of different partial moments. They are well-suited to handle equations where collisions of particles are modelled with a Laplace-Beltrami operator. We generalize the concept of mixed moments to two dimensions. In the context of minimum-entropy models, the resulting hyperbolic system of equations has desirable properties (entropy-diminishing, bounded eigenvalues), removing some drawbacks of the well-known M1 model. We furthermore provide a realizability theory for a first-order system of mixed moments by linking it to the corresponding quarter-moment theory. Additionally, we derive a type of Kershaw closures for mixed-and quarter-moment models, giving an efficient closure (compared to minimum-entropy models). The derived closures are investigated for different benchmark problems.

Keywords: Radiation transport, moment models, realizability, mixed moments, Laplace-Beltrami operator.
Mathematics Subject Classification: Primary: 35L40, 35Q84, 65M08, 65M70.
Citation: Florian Schneider, Andreas Roth, Jochen Kall. First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions. Kinetic & Related Models, 2017, 10 (4) : 1127-1161. doi: 10.3934/krm.2017044
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), 489-508.  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), B361-B391.  doi: 10.1137/11084772X.  Google Scholar

[3]

G. W. Alldredge and F. Schneider, A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, Journal of Computational Physics, 295 (2015), 665-684.  doi: 10.1016/j.jcp.2015.04.034.  Google Scholar

[4]

U. Ascher, S. Ruuth and R. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, 25 (1997), 151-167.  doi: 10.1016/S0168-9274(97)00056-1.  Google Scholar

[5]

G. I. Bell and S. Glasstone, Nuclear Reactor Theory Technical report, Division of Technical Information, US Atomic Energy Commission, 1970. Google Scholar

[6]

M. A. Blanco, M. Flórez and M. Bermejo, Evaluation of the rotation matrices in the basis of real spherical harmonics, Journal of Molecular Structure, 419 (1997), 19-27.  doi: 10.1016/S0166-1280(97)00185-1.  Google Scholar

[7]

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), 543-566.  doi: 10.1016/S0022-4073(00)00099-6.  Google Scholar

[8]

T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, Journal of Computational Physics, 210 (2005), 386-399.  doi: 10.1016/j.jcp.2005.04.011.  Google Scholar

[9]

J. A. Carrillo, A. Klar and A. Roth, Single to double mill small noise transition via semi-lagrangian finite volume methods, Commun. Math. Sci., 14 (2016), 1111-1136.  doi: 10.4310/CMS.2016.v14.n4.a12.  Google Scholar

[10]

R. E. R. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math, 17 (1991), 603–635, URL https://www.math.uh.edu/~hjm/v017n4/ 0603CURTO.pdf  Google Scholar

[11]

B. Dubroca and J.-L. Feugeas, Entropic moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.  Google Scholar

[12]

B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584-596.  doi: 10.1006/jcph.2002.7106.  Google Scholar

[13]

A. S. Eddington, The Internal Constitution of the Stars Dover, 1926. Google Scholar

[14]

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements Applied Mathematical Sciences, Springer New York, 2004, https://books.google.de/books?id=CCjm79FbJbcC. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[15]

G. D. Fies and M. Vianello, Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102–112, URL http://www.emis. ams.org/journals/ETNA/vol.39.2012/pp102-112.dir/pp102-112.pdf. Google Scholar

[16]

G. D. Fies and M. Vianello, Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102-112.   Google Scholar

[17]

M. Frank, Partial moment entropy approximation to radiative heat transfer, Pamm, 5 (2005), 659-660.  doi: 10.1002/pamm.200510306.  Google Scholar

[18]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1-18.  doi: 10.1016/j.jcp.2006.01.038.  Google Scholar

[19]

M. Frank, H. Hensel and A. Klar, A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy, SIAM Journal on Applied Mathematics, 67 (2007), 582-603.  doi: 10.1137/06065547X.  Google Scholar

[20]

W. Fulton, Eigenvalues of sums of Hermitian matrices, Séminaire Bourbaki, 40 (1998), 255-269.   Google Scholar

[21]

C. K. Garrett and C. D. Hauck, A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theory and Statistical Physics, 42 (2013), 203-235.  doi: 10.1080/00411450.2014.910226.  Google Scholar

[22]

E. M. Gelbard, Simplified Spherical Harmonics Equations and Their Use in Shielding Problems Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. Google Scholar

[23]

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.  Google Scholar

[24]

C. D. Hauck, M. Frank and E. Olbrant, Perturbed, entropy-based closure for radiative transfer, SIAM Journal on Applied Mathematics, 6 (2013), 557-587.  doi: 10.3934/krm.2013.6.557.  Google Scholar

[25]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, to appear in Phys. Med. Biol., 51 (2006), 675-693.  doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[26]

J. H. Jeans, The equations of radiative transfer of energy, Monthly Notices Royal Astronomical Society, 78 (1917), 28-36.  doi: 10.1093/mnras/78.1.28.  Google Scholar

[27]

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

[28]

C. Kelley, Solving Nonlinear Equations with Newton's Method Society for Industrial and Applied Mathematics, 2003. doi: 10.1137/1. 9780898718898.  Google Scholar

[29]

D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation, http://www.osti.gov/bridge/product.biblio.jsp?osti_id=104974. Google Scholar

[30]

C. J. Knight and A. C. R. Newbery, Trigonometric and Gaussian quadrature, Mathematics of Computation, 24 (1970), 575-581.  doi: 10.1090/S0025-5718-1970-0275672-4.  Google Scholar

[31]

V. I. Lebedev and D. N. Laikov, A quadrature formula for the sphere of the 131st algebraic order of accuracy, in Doklady. Mathematics, vol. 59, MAIK Nauka/Interperiodica, 1999,477– 481. Google Scholar

[32]

C. D. Levermore, Relating eddington factors to flux limiters, Journal of Quantitative Spectroscopy and Radiative Transfer, 31 (1984), 149-160.  doi: 10.1016/0022-4073(84)90112-2.  Google Scholar

[33]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[34]

W. R. Martin, The application of the finite element method to the neutron transport equation, 1–232. Google Scholar

[35]

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

[36]

P. Monreal, Moment Realizability and Kershaw Closures in Radiative Transfer PhD thesis, TU Aachen, 2012. Google Scholar

[37]

P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer, arXiv preprint, arXiv: 0812.3063, 1–18, URL http://arxiv.org/abs/0812.3063. Google Scholar

[38]

G. C. Pomraning, The Fokker-Planck operator as an asymptotic limit, Math. Mod. Meth. Appl. Sci., 2 (1992), 21-36.  doi: 10.1142/S021820259200003X.  Google Scholar

[39]

A. Roth, Numerical Schemes for Kinetic Equations with Applications to Fibre Lay-Down and Interacting Particles Verlag Dr. Hut, 2014. Google Scholar

[40]

A. Roth, A. Klar, B. Simeon and E. Zharovsky, A semi-lagrangian method for 3-d fokker planck equations for stochastic dynamical systems on the sphere, Journal of Scientific Computing, 61 (2014), 513-532.  doi: 10.1007/s10915-014-9835-z.  Google Scholar

[41]

F. Schneider, First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions: Code 2016. doi: 10.5281/zenodo.48753.  Google Scholar

[42]

F. Schneider, Implicit-explicit, realizability-preserving first-order scheme for moment models with Lipschitz-continuous source terms, arXiv preprint, http://arxiv.org/abs/1611.01314. Google Scholar

[43]

F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅰ: Model derivation, Journal of Computational Physics, 322 (2016), 905-919.  doi: 10.1016/j.jcp.2016.02.080.  Google Scholar

[44]

F. Schneider, Moment Models in Radiation Transport Equations Dr. Hut Verlag, 2016. Google Scholar

[45]

F. Schneider, G. W. Alldredge, M. Frank and A. Klar, Higher order mixed-moment approximations for the Fokker--Planck equation in one space dimension, SIAM Journal on Applied Mathematics, 74 (2014), 1087-1114.  doi: 10.1137/130934210.  Google Scholar

[46]

F. Schneider, G. W. Alldredge and J. Kall, A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kinetic and Related Models, 9 (2016), 193-215.  doi: 10.3934/krm.2016.9.193.  Google Scholar

[47]

B. Seibold and M. Frank, StaRMAP—A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer, ACM Transactions on Mathematical Software, 41 (2014), 1-28.  doi: 10.1145/2590808.  Google Scholar

[48]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Mathematische Annalen, 71 (1912), 441-479.  doi: 10.1007/BF01456804.  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), 489-508.  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), B361-B391.  doi: 10.1137/11084772X.  Google Scholar

[3]

G. W. Alldredge and F. Schneider, A realizability-preserving discontinuous Galerkin scheme for entropy-based moment closures for linear kinetic equations in one space dimension, Journal of Computational Physics, 295 (2015), 665-684.  doi: 10.1016/j.jcp.2015.04.034.  Google Scholar

[4]

U. Ascher, S. Ruuth and R. Spiteri, Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations, Applied Numerical Mathematics, 25 (1997), 151-167.  doi: 10.1016/S0168-9274(97)00056-1.  Google Scholar

[5]

G. I. Bell and S. Glasstone, Nuclear Reactor Theory Technical report, Division of Technical Information, US Atomic Energy Commission, 1970. Google Scholar

[6]

M. A. Blanco, M. Flórez and M. Bermejo, Evaluation of the rotation matrices in the basis of real spherical harmonics, Journal of Molecular Structure, 419 (1997), 19-27.  doi: 10.1016/S0166-1280(97)00185-1.  Google Scholar

[7]

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), 543-566.  doi: 10.1016/S0022-4073(00)00099-6.  Google Scholar

[8]

T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, Journal of Computational Physics, 210 (2005), 386-399.  doi: 10.1016/j.jcp.2005.04.011.  Google Scholar

[9]

J. A. Carrillo, A. Klar and A. Roth, Single to double mill small noise transition via semi-lagrangian finite volume methods, Commun. Math. Sci., 14 (2016), 1111-1136.  doi: 10.4310/CMS.2016.v14.n4.a12.  Google Scholar

[10]

R. E. R. Curto and L. A. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Math, 17 (1991), 603–635, URL https://www.math.uh.edu/~hjm/v017n4/ 0603CURTO.pdf  Google Scholar

[11]

B. Dubroca and J.-L. Feugeas, Entropic moment closure hierarchy for the radiative transfer equation, C. R. Acad. Sci. Paris Ser. I, 329 (1999), 915-920.  doi: 10.1016/S0764-4442(00)87499-6.  Google Scholar

[12]

B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584-596.  doi: 10.1006/jcph.2002.7106.  Google Scholar

[13]

A. S. Eddington, The Internal Constitution of the Stars Dover, 1926. Google Scholar

[14]

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements Applied Mathematical Sciences, Springer New York, 2004, https://books.google.de/books?id=CCjm79FbJbcC. doi: 10.1007/978-1-4757-4355-5.  Google Scholar

[15]

G. D. Fies and M. Vianello, Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102–112, URL http://www.emis. ams.org/journals/ETNA/vol.39.2012/pp102-112.dir/pp102-112.pdf. Google Scholar

[16]

G. D. Fies and M. Vianello, Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012), 102-112.   Google Scholar

[17]

M. Frank, Partial moment entropy approximation to radiative heat transfer, Pamm, 5 (2005), 659-660.  doi: 10.1002/pamm.200510306.  Google Scholar

[18]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1-18.  doi: 10.1016/j.jcp.2006.01.038.  Google Scholar

[19]

M. Frank, H. Hensel and A. Klar, A fast and accurate moment method for the Fokker-Planck equation and applications to electron radiotherapy, SIAM Journal on Applied Mathematics, 67 (2007), 582-603.  doi: 10.1137/06065547X.  Google Scholar

[20]

W. Fulton, Eigenvalues of sums of Hermitian matrices, Séminaire Bourbaki, 40 (1998), 255-269.   Google Scholar

[21]

C. K. Garrett and C. D. Hauck, A comparison of moment closures for linear kinetic transport equations: The line source benchmark, Transport Theory and Statistical Physics, 42 (2013), 203-235.  doi: 10.1080/00411450.2014.910226.  Google Scholar

[22]

E. M. Gelbard, Simplified Spherical Harmonics Equations and Their Use in Shielding Problems Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. Google Scholar

[23]

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.  Google Scholar

[24]

C. D. Hauck, M. Frank and E. Olbrant, Perturbed, entropy-based closure for radiative transfer, SIAM Journal on Applied Mathematics, 6 (2013), 557-587.  doi: 10.3934/krm.2013.6.557.  Google Scholar

[25]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, to appear in Phys. Med. Biol., 51 (2006), 675-693.  doi: 10.1088/0031-9155/51/3/013.  Google Scholar

[26]

J. H. Jeans, The equations of radiative transfer of energy, Monthly Notices Royal Astronomical Society, 78 (1917), 28-36.  doi: 10.1093/mnras/78.1.28.  Google Scholar

[27]

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

[28]

C. Kelley, Solving Nonlinear Equations with Newton's Method Society for Industrial and Applied Mathematics, 2003. doi: 10.1137/1. 9780898718898.  Google Scholar

[29]

D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation, http://www.osti.gov/bridge/product.biblio.jsp?osti_id=104974. Google Scholar

[30]

C. J. Knight and A. C. R. Newbery, Trigonometric and Gaussian quadrature, Mathematics of Computation, 24 (1970), 575-581.  doi: 10.1090/S0025-5718-1970-0275672-4.  Google Scholar

[31]

V. I. Lebedev and D. N. Laikov, A quadrature formula for the sphere of the 131st algebraic order of accuracy, in Doklady. Mathematics, vol. 59, MAIK Nauka/Interperiodica, 1999,477– 481. Google Scholar

[32]

C. D. Levermore, Relating eddington factors to flux limiters, Journal of Quantitative Spectroscopy and Radiative Transfer, 31 (1984), 149-160.  doi: 10.1016/0022-4073(84)90112-2.  Google Scholar

[33]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021-1065.  doi: 10.1007/BF02179552.  Google Scholar

[34]

W. R. Martin, The application of the finite element method to the neutron transport equation, 1–232. Google Scholar

[35]

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

[36]

P. Monreal, Moment Realizability and Kershaw Closures in Radiative Transfer PhD thesis, TU Aachen, 2012. Google Scholar

[37]

P. Monreal and M. Frank, Higher order minimum entropy approximations in radiative transfer, arXiv preprint, arXiv: 0812.3063, 1–18, URL http://arxiv.org/abs/0812.3063. Google Scholar

[38]

G. C. Pomraning, The Fokker-Planck operator as an asymptotic limit, Math. Mod. Meth. Appl. Sci., 2 (1992), 21-36.  doi: 10.1142/S021820259200003X.  Google Scholar

[39]

A. Roth, Numerical Schemes for Kinetic Equations with Applications to Fibre Lay-Down and Interacting Particles Verlag Dr. Hut, 2014. Google Scholar

[40]

A. Roth, A. Klar, B. Simeon and E. Zharovsky, A semi-lagrangian method for 3-d fokker planck equations for stochastic dynamical systems on the sphere, Journal of Scientific Computing, 61 (2014), 513-532.  doi: 10.1007/s10915-014-9835-z.  Google Scholar

[41]

F. Schneider, First-order quarter-and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions: Code 2016. doi: 10.5281/zenodo.48753.  Google Scholar

[42]

F. Schneider, Implicit-explicit, realizability-preserving first-order scheme for moment models with Lipschitz-continuous source terms, arXiv preprint, http://arxiv.org/abs/1611.01314. Google Scholar

[43]

F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅰ: Model derivation, Journal of Computational Physics, 322 (2016), 905-919.  doi: 10.1016/j.jcp.2016.02.080.  Google Scholar

[44]

F. Schneider, Moment Models in Radiation Transport Equations Dr. Hut Verlag, 2016. Google Scholar

[45]

F. Schneider, G. W. Alldredge, M. Frank and A. Klar, Higher order mixed-moment approximations for the Fokker--Planck equation in one space dimension, SIAM Journal on Applied Mathematics, 74 (2014), 1087-1114.  doi: 10.1137/130934210.  Google Scholar

[46]

F. Schneider, G. W. Alldredge and J. Kall, A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry, Kinetic and Related Models, 9 (2016), 193-215.  doi: 10.3934/krm.2016.9.193.  Google Scholar

[47]

B. Seibold and M. Frank, StaRMAP—A Second Order Staggered Grid Method for Spherical Harmonics Moment Equations of Radiative Transfer, ACM Transactions on Mathematical Software, 41 (2014), 1-28.  doi: 10.1145/2590808.  Google Scholar

[48]

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Mathematische Annalen, 71 (1912), 441-479.  doi: 10.1007/BF01456804.  Google Scholar

Figure 1.  Interpolation between two realizability boundaries in the projected three-space $(\phi _{\mathcal{S}_x^ + }^{\left( {1,0} \right)},\phi _{\mathcal{S}_x^ - }^{\left( {1,0} \right)},\phi _{\mathcal{S}_y^ + }^{\left( {0,1} \right)})$ along an isoline of $\mathcal{N}\left( {{\phi ^{\left| 1 \right|}}} \right)$. The realizable set with respect to the quadrant ${\mathcal{S}_{ - + }}$ is plotted in grey
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Figure 2.  The eigenvector ${\bf{v}}_1({\phi _{{\mathcal{S}_{ + + }}}})$ (blue arrows) of the ${\rm{Q}}{{\rm{M}}_1}$ second moment $\phi _{{\mathcal{S}_{ + + }}}^{\left| 2 \right|}$ (Figure (a)) and its minimal deviation from $\phi _{{\mathcal{S}_{+ +}}}^{\left| 1 \right|}$ in degrees (Figure (b))
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Figure 3.  Components of the $Q{K_1}$ second moment on ${\mathcal{S}_{ + + }}$
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Figure 4.  Deviation of $\phi _{{\mathcal{S}_{ + + }}}^{\left( {2,0} \right)}$ and $\phi _{{\mathcal{S}_{ + + }}}^{\left( {2,0} \right)}$ for ${\rm{Q}}{{\rm{M}}_1}$ and $Q{K_1}$, respectively
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Figure 5.  Eigenvalues of the flux Jacobian in $x$-direction ${\bf{F}}_1^\prime$
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Figure 6.  Minimal and maximal distance of adjacent eigenvalues of the flux Jacobian ${\bf{F}}_1^\prime$
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Figure 7.  Visualization of the coefficients $\rho _{\iota \pm}$ for the example $\phi _{{\mathcal{S}_{+ +}}}^{\left| 1 \right|}=\left(0.4,0.6\right)^T$. The length of the arrows represent the maximal value for the respective coefficient
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Figure 8.  Second moment $\phi _{\mathcal{S}_y^ + }^{\left( {0,2} \right)}$ for the ${\rm{M}}{{\rm{M}}_1}$ and Kershaw model for $\phi _{\mathcal{S}_y^ + }^{\left( {0,1} \right)} = -\phi _{\mathcal{S}_y^ - }^{\left( {0,1} \right)} = \frac14$
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Figure 9.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1.2$ in the two-beams test case
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Figure 10.  Local particle density ${u^{\left( {0,0} \right)}}$ at $t = 1.2$ in the opposing two-beams test case
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Figure 11.  Local particle density ${u^{\left( {0,0} \right)}}$ at $t = 1.2$ in the rotated two-beams test case
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Figure 12.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1$ in the source-beam test case
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Figure 13.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 2.5$ in the source-beam test case
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Figure 14.  Horizontal cut ($y = 1.5$) of the local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1$ and $t = 2.5$, respectively, in the source-beam test case
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Figure 15.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 0.45$ in the line-source test case
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Figure 16.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 0.45$ in the line-source test case, horizontal and diagonal cuts
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Table 1.  Absolute and relative running times (wrt. ${\rm{M}}{{\rm{M}}_1}$) of some models for the source-beam test
Model $n$ Abs. runtime [s] Rel. runtime
${\rm{M}}{{\rm{M}}_1}$ 5 1.305e+05 3.885e-01
${\rm{M}}{{\rm{M}}_2}$ 13 3.359e+05 1.0
${\rm{M}}{{\rm{K}}_1}$ 5 3.408e+01 1.015e-04
${{\rm{M}}_1}$ 3 9.418e+04 2.804e-01
${{\rm{M}}_2}$ 6 2.269e+05 6.757e-01
$Q{K_1}$ 12 5.309e+01 1.581e-04
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Model $n$ Abs. runtime [s] Rel. runtime
${\rm{M}}{{\rm{M}}_1}$ 5 1.305e+05 3.885e-01
${\rm{M}}{{\rm{M}}_2}$ 13 3.359e+05 1.0
${\rm{M}}{{\rm{K}}_1}$ 5 3.408e+01 1.015e-04
${{\rm{M}}_1}$ 3 9.418e+04 2.804e-01
${{\rm{M}}_2}$ 6 2.269e+05 6.757e-01
$Q{K_1}$ 12 5.309e+01 1.581e-04
Table 2.  Absolute and relative running times (wrt. ${\rm{M}}{{\rm{M}}_1}$) of some models for the source-beam test.
Model $n$ Abs. runtime [s] Rel. runtime
${\rm{M}}{{\rm{M}}_1}$ 5 2.489e+05 1.0
${\rm{M}}{{\rm{M}}_2}$ 13 9.526e+04 3.828e-01
${\rm{M}}{{\rm{K}}_1}$ poly 5 1.535e+02 6.169e-04
${{\rm{M}}_1}$ 3 6.639e+04 2.668e-01
${{\rm{M}}_2}$ 6 1.189e+05 4.777e-01
${{\rm{M}}_3}$ 12 2.232e+05 8.967e-01
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Model $n$ Abs. runtime [s] Rel. runtime
${\rm{M}}{{\rm{M}}_1}$ 5 2.489e+05 1.0
${\rm{M}}{{\rm{M}}_2}$ 13 9.526e+04 3.828e-01
${\rm{M}}{{\rm{K}}_1}$ poly 5 1.535e+02 6.169e-04
${{\rm{M}}_1}$ 3 6.639e+04 2.668e-01
${{\rm{M}}_2}$ 6 1.189e+05 4.777e-01
${{\rm{M}}_3}$ 12 2.232e+05 8.967e-01
Table 3.  Absolute and relative running times (wrt. ${\rm{M}}{{\rm{M}}_3}$) of some models for the line-source test
Model $n$ Abs. runtime [s] Rel. runtime
${\rm{M}}{{\rm{M}}_1}$ 5 1.638e+05 2.091e-01
${\rm{M}}{{\rm{M}}_2}$ 13 3.491e+05 4.456e-01
${\rm{M}}{{\rm{M}}_3}$ 25 7.834e+05 1.0
${\rm{M}}{{\rm{K}}_1}$ poly 5 1.888e+02 2.409e-04
${\rm{M}}{{\rm{K}}_1}$ 5 2.898e+04 3.700e-02
${{\rm{M}}_1}$ 3 5.458e+04 6.968e-02
${{\rm{M}}_2}$ 6 9.696e+04 1.238e-01
${{\rm{M}}_3}$ 10 1.823e+05 2.327e-01
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Model $n$ Abs. runtime [s] Rel. runtime
${\rm{M}}{{\rm{M}}_1}$ 5 1.638e+05 2.091e-01
${\rm{M}}{{\rm{M}}_2}$ 13 3.491e+05 4.456e-01
${\rm{M}}{{\rm{M}}_3}$ 25 7.834e+05 1.0
${\rm{M}}{{\rm{K}}_1}$ poly 5 1.888e+02 2.409e-04
${\rm{M}}{{\rm{K}}_1}$ 5 2.898e+04 3.700e-02
${{\rm{M}}_1}$ 3 5.458e+04 6.968e-02
${{\rm{M}}_2}$ 6 9.696e+04 1.238e-01
${{\rm{M}}_3}$ 10 1.823e+05 2.327e-01
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