
Left:
Right:
Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schrödinger-Str., 67663 Kaiserslautern, Germany |
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.
[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] |
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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] |
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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.
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[16] |
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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. Google Scholar |
[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] |
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[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. Google Scholar |
[29] |
MATLAB, version 9. 1. 0. 441655 (R2016b), The MathWorks Inc., Natick, Massachusetts, 2015. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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
[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. Google Scholar |
[14] |
I. Filippidis, fig2u3d, URL https://de.mathworks.com/matlabcentral/fileexchange/37640-export-figure-to-3d-interactive-pdf. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[29] |
MATLAB, version 9. 1. 0. 441655 (R2016b), The MathWorks Inc., Natick, Massachusetts, 2015. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. |
|
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|
| |
| 0.0260 | 1.9058 | 0.3287 | 1.0283 | -0.9948 | -0.1326 | 0.1326 | 0.9948 |
| 0.0260 | 1.9058 | 0.0438 | 1.7811 | -0.9948 | -0.3778 | 0.3778 | 0.9948 |
| 0.0314 | 1.2696 | 0.2066 | 0.5884 | -0.9943 | -0.1326 | 0.1326 | 0.4376 |
| 0.0314 | 1.2696 | 0.2066 | 0.5884 | -0.4376 | -0.1326 | 0.1326 | 0.9943 |
| 0.0392 | 1.9058 | 0.3949 | 0.8791 | -0.9921 | -0.1326 | 0.1326 | 0.9921 |
|
|
|
|
|
|
|
| |
| 0.0260 | 1.9058 | 0.3287 | 1.0283 | -0.9948 | -0.1326 | 0.1326 | 0.9948 |
| 0.0260 | 1.9058 | 0.0438 | 1.7811 | -0.9948 | -0.3778 | 0.3778 | 0.9948 |
| 0.0314 | 1.2696 | 0.2066 | 0.5884 | -0.9943 | -0.1326 | 0.1326 | 0.4376 |
| 0.0314 | 1.2696 | 0.2066 | 0.5884 | -0.4376 | -0.1326 | 0.1326 | 0.9943 |
| 0.0392 | 1.9058 | 0.3949 | 0.8791 | -0.9921 | -0.1326 | 0.1326 | 0.9921 |
|
|
|
|
| | | | |
| 0.0334 | 1.8812 | 0.1921 | 1.0945 | -0.9936 | -0.2315 | 0.2315 | 0.9936 |
0.0334 | 1.8812 | 0.0393 | 1.8611 | -0.9936 | -0.9709 | 0.9709 | 0.9936 | |
| 0.0603 | 1.4064 | 0.2291 | 0.8726 | -0.9917 | -0.2806 | 0.2315 | 0.9855 |
0.0603 | 1.4064 | 0.2291 | 0.8726 | -0.9855 | -0.2315 | 0.2806 | 0.9917 | |
| 0.0527 | 1.8138 | 0.1792 | 1.0526 | -0.9897 | -0.2789 | 0.2789 | 0.9897 |
|
|
|
|
| | | | |
| 0.0334 | 1.8812 | 0.1921 | 1.0945 | -0.9936 | -0.2315 | 0.2315 | 0.9936 |
0.0334 | 1.8812 | 0.0393 | 1.8611 | -0.9936 | -0.9709 | 0.9709 | 0.9936 | |
| 0.0603 | 1.4064 | 0.2291 | 0.8726 | -0.9917 | -0.2806 | 0.2315 | 0.9855 |
0.0603 | 1.4064 | 0.2291 | 0.8726 | -0.9855 | -0.2315 | 0.2806 | 0.9917 | |
| 0.0527 | 1.8138 | 0.1792 | 1.0526 | -0.9897 | -0.2789 | 0.2789 | 0.9897 |
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José Antonio Alcántara, Simone Calogero. On a relativistic Fokker-Planck equation in kinetic theory. Kinetic & Related Models, 2011, 4 (2) : 401-426. doi: 10.3934/krm.2011.4.401 |
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Helge Dietert, Josephine Evans, Thomas Holding. Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus. Kinetic & Related Models, 2018, 11 (6) : 1427-1441. doi: 10.3934/krm.2018056 |
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Andreas Denner, Oliver Junge, Daniel Matthes. Computing coherent sets using the Fokker-Planck equation. Journal of Computational Dynamics, 2016, 3 (2) : 163-177. doi: 10.3934/jcd.2016008 |
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Ioannis Markou. Hydrodynamic limit for a Fokker-Planck equation with coefficients in Sobolev spaces. Networks & Heterogeneous Media, 2017, 12 (4) : 683-705. doi: 10.3934/nhm.2017028 |
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Manh Hong Duong, Yulong Lu. An operator splitting scheme for the fractional kinetic Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5707-5727. doi: 10.3934/dcds.2019250 |
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Giuseppe Toscani. A Rosenau-type approach to the approximation of the linear Fokker-Planck equation. Kinetic & Related Models, 2018, 11 (4) : 697-714. doi: 10.3934/krm.2018028 |
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John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete & Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371 |
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Martin Burger, Ina Humpert, Jan-Frederik Pietschmann. On Fokker-Planck equations with In- and Outflow of Mass. Kinetic & Related Models, 2020, 13 (2) : 249-277. doi: 10.3934/krm.2020009 |
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Fabio Camilli, Serikbolsyn Duisembay, Qing Tang. Approximation of an optimal control problem for the time-fractional Fokker-Planck equation. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021013 |
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Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65 |
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Joseph G. Conlon, André Schlichting. A non-local problem for the Fokker-Planck equation related to the Becker-Döring model. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 1821-1889. doi: 10.3934/dcds.2019079 |
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Simon Plazotta. A BDF2-approach for the non-linear Fokker-Planck equation. Discrete & Continuous Dynamical Systems, 2019, 39 (5) : 2893-2913. doi: 10.3934/dcds.2019120 |
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Patrick Cattiaux, Elissar Nasreddine, Marjolaine Puel. Diffusion limit for kinetic Fokker-Planck equation with heavy tails equilibria: The critical case. Kinetic & Related Models, 2019, 12 (4) : 727-748. doi: 10.3934/krm.2019028 |
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Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic & Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011 |
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