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A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry
1. | Fachbereich Mathematik, TU Kaiserslautern, Erwin-Schrödinger-Str, 67663 Kaiserslautern, Germany, Germany |
2. | Department of Mathematics, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany |
References:
[1] |
, http://www.mathematik.uni-kl.de/techno/research/high-order/weno/,, URL , ().
|
[2] |
G. Alldredge, C. Hauck and A. 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. |
[3] |
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. |
[4] |
G. 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. |
[5] |
C. Bresten, S. Gottlieb, Z. Grant, D. Higgs, D. I. Ketcheson and A. Németh, Strong stability preserving multistep Runge-Kutta methods,, URL , ().
|
[6] |
T. A. Brunner, Forms of Approximate Radiation Transport,, Tech. Rep SAND2002-1778., (): 2002.
|
[7] |
J.-B. Cheng, E. F. Toro, S. Jiang and W. Tang, A sub-cell weno reconstruction method for spatial derivatives in the ader scheme, Journal of Computational Physics, 251 (2013), 53-80.
doi: 10.1016/j.jcp.2013.05.034. |
[8] |
B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., 52 (1989), 411-435.
doi: 10.2307/2008474. |
[9] |
R. Curto and L. Fialkow, Recursiveness, positivity and truncated moment problems, Houston J. Math, 17 (1991), 603-635. |
[10] |
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. |
[11] |
B. Dubroca, M. Frank, A. Klar and G. Thömmes, Half space moment approximation to the radiative heat transfer equations, ZAMM, 83 (2003), 853-858.
doi: 10.1002/zamm.200310055. |
[12] |
B. Dubroca and A. Klar, Half moment closure for radiative transfer equations, J. Comput. Phys., 180 (2002), 584-596.
doi: 10.1006/jcph.2002.7106. |
[13] |
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. |
[14] |
M. Frank, C. D. Hauck and E. Olbrant, Perturbed, entropy-based closure for radiative transfer, Kinetic and Related Models, 6 (2013), 557-587.
doi: 10.3934/krm.2013.6.557. |
[15] |
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. |
[16] |
E. M. Gelbard, Simplified Spherical Harmonics Equations and Their Use in Shielding Problems, Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. |
[17] |
S. Gottlieb, On high order strong stability preserving Runge-Kutta and multi step time discretizations, Journal of Scientific Computing, 25 (2005), 105-128.
doi: 10.1007/s10915-004-4635-5. |
[18] |
C. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 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. |
[19] |
M. Junk, Maximum entropy for reduced moment problems, Math. Meth. Mod. Appl. Sci., 10 (2000), 1001-1025.
doi: 10.1142/S0218202500000513. |
[20] |
D. Ketcheson, S. Gottlieb and C. Macdonald, Strong stability preserving two-step Runge-Kutta methods, SIAM Journal on Numerical Analysis, 49 (2011), 2618-2639.
doi: 10.1137/10080960X. |
[21] |
C. K. S. Lam and C. P. T. Groth, Numerical prediction of three-dimensional non-equilibrium flows using the regularized gaussian moment closure, in Proc. of the 20th AIAA Computational Fluid Dynamics Conference, 3401 (2011), 26pp.
doi: 10.2514/6.2014-0229. |
[22] |
E. W. Larsen and C. G. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry-I: Analysis, Nucl. Sci. Eng, 109 (1991), 49-75. |
[23] |
E. W. Larsen and C. G. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry-II: Numerical results, Nucl. Sci. Eng., 109 (1991), 76-85. |
[24] |
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[25] |
E. E. Lewis and W. F. Miller Jr., Computational Methods in Neutron Transport, John Wiley and Sons, New York, 1984. |
[26] |
R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical harmonics expansions for radiative transfer, Journal of Computational Physics, 229 (2010), 5597-5614, URL http://www.sciencedirect.com/science/article/pii/S0021999110001622.
doi: 10.1016/j.jcp.2010.03.043. |
[27] |
J. G. McDonald and C. P. T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Continuum Mechanics and Thermodynamics, 25 (2013), 573-603.
doi: 10.1007/s00161-012-0252-y. |
[28] |
E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer, Journal of Computational Physics, 231 (2012), 5612-5639, URL http://linkinghub.elsevier.com/retrieve/pii/S0021999112001362.
doi: 10.1016/j.jcp.2012.03.002. |
[29] |
G. C. Pomraning, Variational boundary conditions for the spherical harmonics approximation to the neutron transport equation, Ann. Phys., 27 (1964), 193-215.
doi: 10.1016/0003-4916(64)90105-8. |
[30] |
D. Radice, E. Abdikamalov, L. Rezzolla and C. D. Ott, A new spherical harmonics scheme for multi-dimensional radiation transport i. static matter configurations, Journal of Computational Physics, 242 (2013), 648-669, URL http://www.sciencedirect.com/science/article/pii/S0021999113001125.
doi: 10.1016/j.jcp.2013.01.048. |
[31] |
S. J. Ruuth and R. J. Spiteri, High-order strong-stability-preserving Runge-Kutta methods with downwind-biased spatial discretizations, SIAM Journal on Numerical Analysis, 42 (2004), 974-996.
doi: 10.1137/S0036142902419284. |
[32] |
F. Schneider, G. 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. |
[33] |
J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society, New York, 1943. |
[34] |
C. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997), 325-432, Lecture Notes in Math., 1697, Springer, Berlin, 1998.
doi: 10.1007/BFb0096355. |
[35] |
H. Struchtrup, Kinetic schemes and boundary conditions for moment equations, Z. Angew. Math. Phys., 51 (2000), 346-365.
doi: 10.1007/s000330050002. |
[36] |
E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, 2009, URL http://books.google.de/books?id=SqEjX0um8o0C.
doi: 10.1007/b79761. |
[37] |
X. Zhang, Maximum-Principle-Satisfying and Positivity-Preserving High Order Schemes for Conservation Laws, PhD thesis, Brown University, 2011, URL www.math.purdue.edu/~zhan1966/thesis.pdf. |
[38] |
X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, Journal of Computational Physics, 229 (2010), 3091-3120, URL http://www.sciencedirect.com/science/article/pii/S0021999109007165.
doi: 10.1016/j.jcp.2009.12.030. |
[39] |
X. Zhang and C.-W. Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, Journal of Computational Physics, 229 (2010), 8918-8934, URL http://www.sciencedirect.com/science/article/pii/S0021999110004535.
doi: 10.1016/j.jcp.2010.08.016. |
[40] |
X. Zhang, Y. Xia and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, Journal of Scientific Computing, 50 (2012), 29-62.
doi: 10.1007/s10915-011-9472-8. |
[41] |
X. Zhang and C. W. Shu, A minimum entropy principle of high order schemes for gas dynamics equations, Numerische Mathematik, 121 (2012), 545-563.
doi: 10.1007/s00211-011-0443-7. |
show all references
References:
[1] |
, http://www.mathematik.uni-kl.de/techno/research/high-order/weno/,, URL , ().
|
[2] |
G. Alldredge, C. Hauck and A. 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. |
[3] |
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. |
[4] |
G. 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. |
[5] |
C. Bresten, S. Gottlieb, Z. Grant, D. Higgs, D. I. Ketcheson and A. Németh, Strong stability preserving multistep Runge-Kutta methods,, URL , ().
|
[6] |
T. A. Brunner, Forms of Approximate Radiation Transport,, Tech. Rep SAND2002-1778., (): 2002.
|
[7] |
J.-B. Cheng, E. F. Toro, S. Jiang and W. Tang, A sub-cell weno reconstruction method for spatial derivatives in the ader scheme, Journal of Computational Physics, 251 (2013), 53-80.
doi: 10.1016/j.jcp.2013.05.034. |
[8] |
B. Cockburn and C.-W. Shu, TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws II: General framework, Math. Comp., 52 (1989), 411-435.
doi: 10.2307/2008474. |
[9] |
R. Curto and L. Fialkow, Recursiveness, positivity and truncated moment problems, Houston J. Math, 17 (1991), 603-635. |
[10] |
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. |
[11] |
B. Dubroca, M. Frank, A. Klar and G. Thömmes, Half space moment approximation to the radiative heat transfer equations, ZAMM, 83 (2003), 853-858.
doi: 10.1002/zamm.200310055. |
[12] |
B. Dubroca and A. Klar, Half moment closure for radiative transfer equations, J. Comput. Phys., 180 (2002), 584-596.
doi: 10.1006/jcph.2002.7106. |
[13] |
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. |
[14] |
M. Frank, C. D. Hauck and E. Olbrant, Perturbed, entropy-based closure for radiative transfer, Kinetic and Related Models, 6 (2013), 557-587.
doi: 10.3934/krm.2013.6.557. |
[15] |
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. |
[16] |
E. M. Gelbard, Simplified Spherical Harmonics Equations and Their Use in Shielding Problems, Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961. |
[17] |
S. Gottlieb, On high order strong stability preserving Runge-Kutta and multi step time discretizations, Journal of Scientific Computing, 25 (2005), 105-128.
doi: 10.1007/s10915-004-4635-5. |
[18] |
C. Hauck, High-order entropy-based closures for linear transport in slab geometry, Commun. Math. Sci., 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. |
[19] |
M. Junk, Maximum entropy for reduced moment problems, Math. Meth. Mod. Appl. Sci., 10 (2000), 1001-1025.
doi: 10.1142/S0218202500000513. |
[20] |
D. Ketcheson, S. Gottlieb and C. Macdonald, Strong stability preserving two-step Runge-Kutta methods, SIAM Journal on Numerical Analysis, 49 (2011), 2618-2639.
doi: 10.1137/10080960X. |
[21] |
C. K. S. Lam and C. P. T. Groth, Numerical prediction of three-dimensional non-equilibrium flows using the regularized gaussian moment closure, in Proc. of the 20th AIAA Computational Fluid Dynamics Conference, 3401 (2011), 26pp.
doi: 10.2514/6.2014-0229. |
[22] |
E. W. Larsen and C. G. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry-I: Analysis, Nucl. Sci. Eng, 109 (1991), 49-75. |
[23] |
E. W. Larsen and C. G. Pomraning, The $P_N$ theory as an asymptotic limit of transport theory in planar geometry-II: Numerical results, Nucl. Sci. Eng., 109 (1991), 76-85. |
[24] |
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), 1021-1065.
doi: 10.1007/BF02179552. |
[25] |
E. E. Lewis and W. F. Miller Jr., Computational Methods in Neutron Transport, John Wiley and Sons, New York, 1984. |
[26] |
R. G. McClarren and C. D. Hauck, Robust and accurate filtered spherical harmonics expansions for radiative transfer, Journal of Computational Physics, 229 (2010), 5597-5614, URL http://www.sciencedirect.com/science/article/pii/S0021999110001622.
doi: 10.1016/j.jcp.2010.03.043. |
[27] |
J. G. McDonald and C. P. T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Continuum Mechanics and Thermodynamics, 25 (2013), 573-603.
doi: 10.1007/s00161-012-0252-y. |
[28] |
E. Olbrant, C. D. Hauck and M. Frank, A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer, Journal of Computational Physics, 231 (2012), 5612-5639, URL http://linkinghub.elsevier.com/retrieve/pii/S0021999112001362.
doi: 10.1016/j.jcp.2012.03.002. |
[29] |
G. C. Pomraning, Variational boundary conditions for the spherical harmonics approximation to the neutron transport equation, Ann. Phys., 27 (1964), 193-215.
doi: 10.1016/0003-4916(64)90105-8. |
[30] |
D. Radice, E. Abdikamalov, L. Rezzolla and C. D. Ott, A new spherical harmonics scheme for multi-dimensional radiation transport i. static matter configurations, Journal of Computational Physics, 242 (2013), 648-669, URL http://www.sciencedirect.com/science/article/pii/S0021999113001125.
doi: 10.1016/j.jcp.2013.01.048. |
[31] |
S. J. Ruuth and R. J. Spiteri, High-order strong-stability-preserving Runge-Kutta methods with downwind-biased spatial discretizations, SIAM Journal on Numerical Analysis, 42 (2004), 974-996.
doi: 10.1137/S0036142902419284. |
[32] |
F. Schneider, G. 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. |
[33] |
J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society, New York, 1943. |
[34] |
C. Shu, Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, Advanced numerical approximation of nonlinear hyperbolic equations (Cetraro, 1997), 325-432, Lecture Notes in Math., 1697, Springer, Berlin, 1998.
doi: 10.1007/BFb0096355. |
[35] |
H. Struchtrup, Kinetic schemes and boundary conditions for moment equations, Z. Angew. Math. Phys., 51 (2000), 346-365.
doi: 10.1007/s000330050002. |
[36] |
E. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, Springer, 2009, URL http://books.google.de/books?id=SqEjX0um8o0C.
doi: 10.1007/b79761. |
[37] |
X. Zhang, Maximum-Principle-Satisfying and Positivity-Preserving High Order Schemes for Conservation Laws, PhD thesis, Brown University, 2011, URL www.math.purdue.edu/~zhan1966/thesis.pdf. |
[38] |
X. Zhang and C.-W. Shu, On maximum-principle-satisfying high order schemes for scalar conservation laws, Journal of Computational Physics, 229 (2010), 3091-3120, URL http://www.sciencedirect.com/science/article/pii/S0021999109007165.
doi: 10.1016/j.jcp.2009.12.030. |
[39] |
X. Zhang and C.-W. Shu, On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, Journal of Computational Physics, 229 (2010), 8918-8934, URL http://www.sciencedirect.com/science/article/pii/S0021999110004535.
doi: 10.1016/j.jcp.2010.08.016. |
[40] |
X. Zhang, Y. Xia and C.-W. Shu, Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, Journal of Scientific Computing, 50 (2012), 29-62.
doi: 10.1007/s10915-011-9472-8. |
[41] |
X. Zhang and C. W. Shu, A minimum entropy principle of high order schemes for gas dynamics equations, Numerische Mathematik, 121 (2012), 545-563.
doi: 10.1007/s00211-011-0443-7. |
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