# American Institute of Mathematical Sciences

March  2016, 9(1): 193-215. doi: 10.3934/krm.2016.9.193

## 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

Received  January 2015 Revised  June 2015 Published  October 2015

We develop a high-order kinetic scheme for entropy-based moment models of a one-dimensional linear kinetic equation in slab geometry. High-order spatial reconstructions are achieved using the weighted essentially non-oscillatory (WENO) method. For time integration we use multi-step Runge-Kutta methods which are strong stability preserving and whose stages and steps can be written as convex combinations of forward Euler steps. We show that the moment vectors stay in the realizable set using these time integrators along with a maximum principle-based kinetic-level limiter, which simultaneously dampens spurious oscillations in the numerical solutions. We present numerical results both on a manufactured solution, where we perform convergence tests showing our scheme has the expected order up to the numerical noise from the optimization routine, as well as on two standard benchmark problems, where we show some of the advantages of high-order solutions and the role of the key parameter in the limiter.
Citation: Florian Schneider, Jochen Kall, Graham 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, 2016, 9 (1) : 193-215. doi: 10.3934/krm.2016.9.193
##### References:
 [1] [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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##### References:
 [1] [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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