American Institute of Mathematical Sciences

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October  2018, 11(5): 1255-1276. doi: 10.3934/krm.2018049

Second-order mixed-moment model with differentiable ansatz function in slab geometry

Florian Schneider

Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany

Received  January 2017 Revised  October 2017 Published  May 2018

Mixed-moment minimum-entropy models (${\rm{M}}{{\rm{M}}_N}$ models) are known to overcome the zero net-flux problem of full-moment minimum entropy ${{\rm{M}}_N}$ models but lack regularity. We study differentiable mixed-moment models (full zeroth and first moment, half higher moments, called ${\rm{DM}}{{\rm{M}}_N}$ models) for a Fokker-Planck equation in one space dimension. Realizability theory for these modification of mixed moments is derived for second order. Numerical tests are performed with a kinetic first-order finite volume scheme and compared with ${{\rm{M}}_N}$, classical ${\rm{M}}{{\rm{M}}_N}$ and a ${{\rm{P}}_N}$ reference scheme.

Keywords: Moment models, minimum entropy, Fokker-Planck equation, realizability, mixed moments.
Mathematics Subject Classification: Primary: 35L40, 35Q84, 65M70; Secondary: 65M08.
Citation: Florian Schneider. Second-order mixed-moment model with differentiable ansatz function in slab geometry. Kinetic and Related Models, 2018, 11 (5) : 1255-1276. doi: 10.3934/krm.2018049
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, URL http://www.sciencedirect.com/science/article/pii/S0021999113007250. doi: 10.1016/j.jcp.2013.10.049.

[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, URL http://epubs.siam.org/doi/abs/10.1137/11084772X. doi: 10.1137/11084772X.

[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, URL http://www.sciencedirect.com/science/article/pii/S0021999115002910. doi: 10.1016/j.jcp.2015.04.034.

[4]

L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekulen, Wien. Ber., 66 (1872), 275–370, URL http://ebooks.cambridge.org/ref/id/CBO9781139381420. doi: 10.1007/978-3-322-84986-1_3.

[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), 543–566, URL http://www.sciencedirect.com/science/article/pii/S0022407300000996. doi: 10.1016/S0022-4073(00)00099-6.

[6]

T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, Journal of Computational Physics, 210 (2005), 386–399, URL http://www.sciencedirect.com/science/article/pii/S0021999105002275. doi: 10.1016/j.jcp.2005.04.011.

[7]

C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, Springer New York, 2012, URL https://books.google.de/books?id=OcTcBwAAQBAJ. doi: 10.1007/978-1-4612-1039-9.

[8]

F. Chalub and P. Markowich, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123–141, URL http://link.springer.com/chapter/10.1007/978-3-7091-0609-9_10. doi: 10.1007/s00605-004-0234-7.

[9]

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

[10]

B. Dubroca, M. Frank, A. Klar and G. Thömmes, Half space moment approximation to the radiative heat transfer equations, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 83 (2003), 853–858. doi: 10.1002/zamm.200310055.

[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.

[12]

B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584–596, URL http://www.sciencedirect.com/science/article/pii/S0021999102971068. doi: 10.1006/jcph.2002.7106.

[13]

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

[14]

I. Filippidis, fig2u3d, URL https://de.mathworks.com/matlabcentral/fileexchange/37640-export-figure-to-3d-interactive-pdf.

[15]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1–18, URL http://www.sciencedirect.com/science/article/pii/S002199910600057X. doi: 10.1016/j.jcp.2006.01.038.

[16]

M. Frank, C. Hauck and E. Olbrant, Perturbed, entropy-based closure for radiative transfer, Kinetic and Related Models, 6 (2013), 557–587, URL http://www.osti.gov/scitech/biblio/1093718. doi: 10.3934/krm.2013.6.557.

[17]

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, URL http://epubs.siam.org/doi/abs/10.1137/06065547X. doi: 10.1137/06065547X.

[18]

E. M. Gelbard, Simplified spherical harmonics equations and their use in shielding problems, Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961.

[19]

K. P. Hadeler, Reaction transport equations in biological modeling, in Mathematical and Computer Modelling, 31 (2000), 75–81. doi: 10.1016/S0895-7177(00)00024-8.

[20]

C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Communications in Mathematical Sciences, 9 (2011), 187–205, URL http://www.ki-net.umd.edu/pubs/files/FRG-2010-Hauck-Cory.entropy{_}kinetic.pdf. doi: 10.4310/CMS.2011.v9.n1.a9.

[21]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, Physics in Medicine and Biology, 51 (2006), 675–693, URL http://www.ncbi.nlm.nih.gov/pubmed/16424588. doi: 10.1088/0031-9155/51/3/013.

[22]

T. Hillen and K. J. Painter, Transport and anisotropic diffusion models for movement in oriented habitats, Lecture Notes in Mathematics, 2071 (2013), 177-222.  doi: 10.1007/978-3-642-35497-7_7.

[23]

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.

[24]

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

[25]

D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation, Lawrence Livermore National Laboratory, UCRL-78378, URL http://www.osti.gov/bridge/product.biblio.jsp?osti{_}id=104974.

[26]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021–1065, URL http://link.springer.com/article/10.1007/BF02179552. doi: 10.1007/BF02179552.

[27]

C. D. Levermore, Moment closure hierarchies for the Boltzmann-Poisson equation, VLSI Design, 6 (1998), 97–101, URL http://www.hindawi.com/journals/vlsi/1998/039370/abs/. doi: 10.1155/1998/39370.

[28]

E. E. Lewis and J. W. F. Miller, Computational Methods in Neutron Transport, John Wiley and Sons, New York, 1984.

[29]

MATLAB, version 9. 1. 0. 441655 (R2016b), The MathWorks Inc., Natick, Massachusetts, 2015.

[30]

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

[31]

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

[32]

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.

[33]

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

[34]

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

[35]

F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅰ: Model derivation, Journal of Computational Physics, 322 (2016), 905–919, URL http://arXiv.org/abs/1511.02714. doi: 10.1016/j.jcp.2016.02.080.

[36]

F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅱ: high-order realizability-preserving discontinuous-Galerkin schemes, Journal of Computational Physics, 322 (2016), 920–935, URL http://arXiv.org/abs/1602.02590. doi: 10.1016/j.jcp.2016.07.014.

[37]

F. Schneider, Moment Models in Radiation Transport Equations, Mathematik edition, Dr. Hut Verlag, 2016.

[38]

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, URL http://epubs.siam.org/doi/abs/10.1137/130934210. doi: 10.1137/130934210.

[39]

F. Schneider, J. Kall and G. Alldredge, 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, URL http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11817. doi: 10.3934/krm.2016.9.193.

[40]

F. Schneider, J. Kall and A. Roth, First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions, Kinetic and Related Models, 10 (2017), 1127–1161, URL http://arXiv.org/abs/1509.02344. doi: 10.3934/krm.2017044.

[41]

D. Wright, M. Frank and A. Klar, The minimum entropy approximation to the radiative transfer equation, Proc. Symp. Appl. Math., 67 (2009), 987-996.  doi: 10.1090/psapm/067.2/2605294.

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, URL http://www.sciencedirect.com/science/article/pii/S0021999113007250. doi: 10.1016/j.jcp.2013.10.049.

[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, URL http://epubs.siam.org/doi/abs/10.1137/11084772X. doi: 10.1137/11084772X.

[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, URL http://www.sciencedirect.com/science/article/pii/S0021999115002910. doi: 10.1016/j.jcp.2015.04.034.

[4]

L. Boltzmann, Weitere Studien über das Wärmegleichgewicht unter Gasmolekulen, Wien. Ber., 66 (1872), 275–370, URL http://ebooks.cambridge.org/ref/id/CBO9781139381420. doi: 10.1007/978-3-322-84986-1_3.

[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), 543–566, URL http://www.sciencedirect.com/science/article/pii/S0022407300000996. doi: 10.1016/S0022-4073(00)00099-6.

[6]

T. A. Brunner and J. P. Holloway, Two-dimensional time dependent Riemann solvers for neutron transport, Journal of Computational Physics, 210 (2005), 386–399, URL http://www.sciencedirect.com/science/article/pii/S0021999105002275. doi: 10.1016/j.jcp.2005.04.011.

[7]

C. Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences, Springer New York, 2012, URL https://books.google.de/books?id=OcTcBwAAQBAJ. doi: 10.1007/978-1-4612-1039-9.

[8]

F. Chalub and P. Markowich, Kinetic models for chemotaxis and their drift-diffusion limits, Monatsh. Math., 142 (2004), 123–141, URL http://link.springer.com/chapter/10.1007/978-3-7091-0609-9_10. doi: 10.1007/s00605-004-0234-7.

[9]

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

[10]

B. Dubroca, M. Frank, A. Klar and G. Thömmes, Half space moment approximation to the radiative heat transfer equations, ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 83 (2003), 853–858. doi: 10.1002/zamm.200310055.

[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.

[12]

B. Dubroca and A. Klar, Half-moment closure for radiative transfer equations, Journal of Computational Physics, 180 (2002), 584–596, URL http://www.sciencedirect.com/science/article/pii/S0021999102971068. doi: 10.1006/jcph.2002.7106.

[13]

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

[14]

I. Filippidis, fig2u3d, URL https://de.mathworks.com/matlabcentral/fileexchange/37640-export-figure-to-3d-interactive-pdf.

[15]

M. Frank, B. Dubroca and A. Klar, Partial moment entropy approximation to radiative heat transfer, Journal of Computational Physics, 218 (2006), 1–18, URL http://www.sciencedirect.com/science/article/pii/S002199910600057X. doi: 10.1016/j.jcp.2006.01.038.

[16]

M. Frank, C. Hauck and E. Olbrant, Perturbed, entropy-based closure for radiative transfer, Kinetic and Related Models, 6 (2013), 557–587, URL http://www.osti.gov/scitech/biblio/1093718. doi: 10.3934/krm.2013.6.557.

[17]

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, URL http://epubs.siam.org/doi/abs/10.1137/06065547X. doi: 10.1137/06065547X.

[18]

E. M. Gelbard, Simplified spherical harmonics equations and their use in shielding problems, Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961.

[19]

K. P. Hadeler, Reaction transport equations in biological modeling, in Mathematical and Computer Modelling, 31 (2000), 75–81. doi: 10.1016/S0895-7177(00)00024-8.

[20]

C. D. Hauck, High-order entropy-based closures for linear transport in slab geometry, Communications in Mathematical Sciences, 9 (2011), 187–205, URL http://www.ki-net.umd.edu/pubs/files/FRG-2010-Hauck-Cory.entropy{_}kinetic.pdf. doi: 10.4310/CMS.2011.v9.n1.a9.

[21]

H. Hensel, R. Iza-Teran and N. Siedow, Deterministic model for dose calculation in photon radiotherapy, Physics in Medicine and Biology, 51 (2006), 675–693, URL http://www.ncbi.nlm.nih.gov/pubmed/16424588. doi: 10.1088/0031-9155/51/3/013.

[22]

T. Hillen and K. J. Painter, Transport and anisotropic diffusion models for movement in oriented habitats, Lecture Notes in Mathematics, 2071 (2013), 177-222.  doi: 10.1007/978-3-642-35497-7_7.

[23]

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.

[24]

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

[25]

D. S. Kershaw, Flux limiting nature's own way: A new method for numerical solution of the transport equation, Lawrence Livermore National Laboratory, UCRL-78378, URL http://www.osti.gov/bridge/product.biblio.jsp?osti{_}id=104974.

[26]

C. D. Levermore, Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996), 1021–1065, URL http://link.springer.com/article/10.1007/BF02179552. doi: 10.1007/BF02179552.

[27]

C. D. Levermore, Moment closure hierarchies for the Boltzmann-Poisson equation, VLSI Design, 6 (1998), 97–101, URL http://www.hindawi.com/journals/vlsi/1998/039370/abs/. doi: 10.1155/1998/39370.

[28]

E. E. Lewis and J. W. F. Miller, Computational Methods in Neutron Transport, John Wiley and Sons, New York, 1984.

[29]

MATLAB, version 9. 1. 0. 441655 (R2016b), The MathWorks Inc., Natick, Massachusetts, 2015.

[30]

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

[31]

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

[32]

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.

[33]

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

[34]

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

[35]

F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅰ: Model derivation, Journal of Computational Physics, 322 (2016), 905–919, URL http://arXiv.org/abs/1511.02714. doi: 10.1016/j.jcp.2016.02.080.

[36]

F. Schneider, Kershaw closures for linear transport equations in slab geometry Ⅱ: high-order realizability-preserving discontinuous-Galerkin schemes, Journal of Computational Physics, 322 (2016), 920–935, URL http://arXiv.org/abs/1602.02590. doi: 10.1016/j.jcp.2016.07.014.

[37]

F. Schneider, Moment Models in Radiation Transport Equations, Mathematik edition, Dr. Hut Verlag, 2016.

[38]

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, URL http://epubs.siam.org/doi/abs/10.1137/130934210. doi: 10.1137/130934210.

[39]

F. Schneider, J. Kall and G. Alldredge, 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, URL http://www.aimsciences.org/journals/displayArticlesnew.jsp?paperID=11817. doi: 10.3934/krm.2016.9.193.

[40]

F. Schneider, J. Kall and A. Roth, First-order quarter- and mixed-moment realizability theory and Kershaw closures for a Fokker-Planck equation in two space dimensions, Kinetic and Related Models, 10 (2017), 1127–1161, URL http://arXiv.org/abs/1509.02344. doi: 10.3934/krm.2017044.

[41]

D. Wright, M. Frank and A. Klar, The minimum entropy approximation to the radiative transfer equation, Proc. Symp. Appl. Math., 67 (2009), 987-996.  doi: 10.1090/psapm/067.2/2605294.

Figure 1.  Two ansatz functions and their derivatives for the ${\rm{M}}{{\rm{M}}_2}$ and ${\rm{DM}}{{\rm{M}}_2}$ model, respectively.
Left: ${\hat \psi }(\mu ) = \exp\left(\left(\mu -\mu ^2\right)1_{[0, 1]}-2\mu 1_{[-1, 0]}\right)$,
Right: ${\hat \psi }(\mu ) = \exp\left(-2\mu -\mu ^21_{[0, 1]}\right)$
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Figure 2.  The normalized realizable set for the differentiable mixed-moment basis of order $N = 2$.
Online version: Press to activate 3D view ($x$-axis (red): ${\phi _{2 + }}$, $y$-axis (green): ${\phi _{2 - }}$, $z$-axis (blue): ${{\phi }_{1}}$)
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Figure 3.  Eigenvalues of the ${\rm{DM}}{{\rm{M}}_2}$ flux Jacobian $\frac{\partial \mathbf{F}(\mathbf{u} )}{\partial \mathbf{u}}$ along ${{\phi }_{1}} = \frac{1}{2}\left({\phi _{2 + }} - \sqrt{{\phi _{2 - }}\, \left(1-{\phi _{2 + }}\right)} +\sqrt{{\phi _{2 + }}\, \left(1-{\phi _{2 - }}\right)} - {\phi _{2 - }}\right)$
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Figure 4.  Minimal and maximal distance between adjacent eigenvalues of the ${\rm{DM}}{{\rm{M}}_2}$ flux Jacobian $\frac{\partial \mathbf{F}(\mathbf{u})}{\partial \mathbf{u}}$ along the cut ${{\phi }_{1}} = \frac{1}{2}\left({\phi _{2 + }} - \sqrt{{\phi _{2 - }}\, \left(1-{\phi _{2 + }}\right)} +\sqrt{{\phi _{2 + }}\, \left(1-{\phi _{2 - }}\right)} - {\phi _{2 - }}\right)$
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Figure 5.  Eigenvalues at $5\%$ regularized boundary moments.
Online version: Press to activate 3D view ($x$-axis (red): ${\phi _{2 + }}$, $y$-axis (green): ${\phi _{2 - }}$, $z$-axis (blue): ${{\phi }_{1}}$)
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Figure 6.  Minimal eigenvalue distance for $5\%$ regularized boundary moments.
Online version: Press to activate 3D view ($x$-axis (red): ${\phi _{2 + }}$, $y$-axis (green): ${\phi _{2 - }}$, $z$-axis (blue): ${{\phi }_{1}}$)
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Figure 7.  Results for the plane-source test at the final time ${t_f} = 1$
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Figure 8.  Results for the source-beam test at the final time ${t_f} = 2.5$
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Figure 9.  DMM2 approximation of the non-differentiable MM2 ansatz (27)
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Table 1.  Quantitative investigation of the minimal and maximal gaps of the eigenvalues at selected parts $S$ of the realizable set for fixed ${{\phi }_{1}} = \frac{1}{2}\left({\phi _{2 + }} - \sqrt{{\phi _{2 - }}\, \left(1-{\phi _{2 + }}\right)} +\sqrt{{\phi _{2 + }}\, \left(1-{\phi _{2 - }}\right)} - {\phi _{2 - }}\right)$
$S$ $\min\limits_{\mathbf{u} \in S} \lambda _{\min}$ $\max\limits_{\mathbf{u} \in S} \lambda _{\max}$ $\mathop {{\rm{mean}}{\lambda _{{\rm{min}}}}}\limits_{\mathbf{u} \in S} $ $\mathop {{\rm{mean}}{\lambda _{{\rm{max}}}}}\limits_{\mathbf{u} \in S} $ $\min\limits_{\mathbf{u} \in S} \lambda _{1}$ $\max\limits_{\mathbf{u} \in S} \lambda _{1}$ $\min\limits_{\mathbf{u} \in S} \lambda _{4}$ $\max\limits_{\mathbf{u} \in S} \lambda _{4}$
$\mathcal{R}_\mathbf{b}{}$0.02601.90580.32871.0283-0.9948-0.13260.13260.9948
${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$0.02601.90580.04381.7811-0.9948-0.37780.37780.9948
${\phi _{2 + }}\leq \frac{1}{120}$0.03141.26960.20660.5884-0.9943-0.13260.13260.4376
${\phi _{2 - }}\leq \frac{1}{120}$0.03141.26960.20660.5884-0.4376-0.13260.13260.9943
${\phi _{2 + }}={\phi _{2 - }}$0.03921.90580.39490.8791-0.9921-0.13260.13260.9921
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$S$ $\min\limits_{\mathbf{u} \in S} \lambda _{\min}$ $\max\limits_{\mathbf{u} \in S} \lambda _{\max}$ $\mathop {{\rm{mean}}{\lambda _{{\rm{min}}}}}\limits_{\mathbf{u} \in S} $ $\mathop {{\rm{mean}}{\lambda _{{\rm{max}}}}}\limits_{\mathbf{u} \in S} $ $\min\limits_{\mathbf{u} \in S} \lambda _{1}$ $\max\limits_{\mathbf{u} \in S} \lambda _{1}$ $\min\limits_{\mathbf{u} \in S} \lambda _{4}$ $\max\limits_{\mathbf{u} \in S} \lambda _{4}$
$\mathcal{R}_\mathbf{b}{}$0.02601.90580.32871.0283-0.9948-0.13260.13260.9948
${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$0.02601.90580.04381.7811-0.9948-0.37780.37780.9948
${\phi _{2 + }}\leq \frac{1}{120}$0.03141.26960.20660.5884-0.9943-0.13260.13260.4376
${\phi _{2 - }}\leq \frac{1}{120}$0.03141.26960.20660.5884-0.4376-0.13260.13260.9943
${\phi _{2 + }}={\phi _{2 - }}$0.03921.90580.39490.8791-0.9921-0.13260.13260.9921
Table 2.  Quantitative investigation of the minimal and maximal gaps of the eigenvalues at selected parts $S$ of the realizable set for the $5\%$-regularized boundary moments
$S$ $\min\limits_{\mathbf{u} \in S} \lambda _{\min}$ $\max\limits_{\mathbf{u} \in S} \lambda _{\max}$ $\mathop {{\rm{mean}}{\lambda _{{\rm{min}}}}}\limits_{\mathbf{u} \in S} $ $\mathop {{\rm{mean}}{\lambda _{{\rm{max}}}}}\limits_{\mathbf{u} \in S} $ $\min\limits_{\mathbf{u} \in S} \lambda _{1}$ $\max\limits_{\mathbf{u} \in S} \lambda _{1}$ $\min\limits_{\mathbf{u} \in S} \lambda _{4}$ $\max\limits_{\mathbf{u} \in S} \lambda _{4}$
$\mathcal{R}_\mathbf{b}{}$0.03341.88120.19211.0945-0.9936-0.23150.23150.9936
${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$0.03341.88120.03931.8611-0.9936-0.97090.97090.9936
${\phi _{2 + }}\leq \frac{1}{120}$0.06031.40640.22910.8726-0.9917-0.28060.23150.9855
${\phi _{2 - }}\leq \frac{1}{120}$0.06031.40640.22910.8726-0.9855-0.23150.28060.9917
${\phi _{2 + }}={\phi _{2 - }}$0.05271.81380.17921.0526-0.9897-0.27890.27890.9897
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$S$ $\min\limits_{\mathbf{u} \in S} \lambda _{\min}$ $\max\limits_{\mathbf{u} \in S} \lambda _{\max}$ $\mathop {{\rm{mean}}{\lambda _{{\rm{min}}}}}\limits_{\mathbf{u} \in S} $ $\mathop {{\rm{mean}}{\lambda _{{\rm{max}}}}}\limits_{\mathbf{u} \in S} $ $\min\limits_{\mathbf{u} \in S} \lambda _{1}$ $\max\limits_{\mathbf{u} \in S} \lambda _{1}$ $\min\limits_{\mathbf{u} \in S} \lambda _{4}$ $\max\limits_{\mathbf{u} \in S} \lambda _{4}$
$\mathcal{R}_\mathbf{b}{}$0.03341.88120.19211.0945-0.9936-0.23150.23150.9936
${\phi _{2 + }}+{\phi _{2 - }}\geq \frac{19}{20}$0.03341.88120.03931.8611-0.9936-0.97090.97090.9936
${\phi _{2 + }}\leq \frac{1}{120}$0.06031.40640.22910.8726-0.9917-0.28060.23150.9855
${\phi _{2 - }}\leq \frac{1}{120}$0.06031.40640.22910.8726-0.9855-0.23150.28060.9917
${\phi _{2 + }}={\phi _{2 - }}$0.05271.81380.17921.0526-0.9897-0.27890.27890.9897
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