\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

  • * Corresponding author: Florian Schneider

    * Corresponding author: Florian Schneider 
Abstract Full Text(HTML) Figure(16) / Table(3) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: Primary: 35L40, 35Q84, 65M08, 65M70.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • 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

    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))

    Figure 3.  Components of the $Q{K_1}$ second moment on ${\mathcal{S}_{ + + }}$

    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

    Figure 5.  Eigenvalues of the flux Jacobian in $x$-direction ${\bf{F}}_1^\prime$

    Figure 6.  Minimal and maximal distance of adjacent eigenvalues of the flux Jacobian ${\bf{F}}_1^\prime$

    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

    Figure 8.  Second moment $\phi _{\mathcal{S}_y^ + }^{\left( {0,2} \right)}$ for the ${\rm{M}}{{\rm{M}}_1}$ and Kershaw model for $\phi _{\mathcal{S}_y^ + }^{\left( {0,1} \right)} = -\phi _{\mathcal{S}_y^ - }^{\left( {0,1} \right)} = \frac14$

    Figure 9.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1.2$ in the two-beams test case

    Figure 10.  Local particle density ${u^{\left( {0,0} \right)}}$ at $t = 1.2$ in the opposing two-beams test case

    Figure 11.  Local particle density ${u^{\left( {0,0} \right)}}$ at $t = 1.2$ in the rotated two-beams test case

    Figure 12.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 1$ in the source-beam test case

    Figure 13.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 2.5$ in the source-beam test case

    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

    Figure 15.  Local particle density ${u^{\left( {0,0} \right)}}$ of some models at $t = 0.45$ in the line-source test case

    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

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
  •   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.
      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.
      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.
      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.
      G. I. Bell and S. Glasstone, Nuclear Reactor Theory Technical report, Division of Technical Information, US Atomic Energy Commission, 1970.
      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.
      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.
      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.
      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.
      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
      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.
      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.
      A. S. Eddington, The Internal Constitution of the Stars Dover, 1926.
      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.
      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.
      G. D. Fies  and  M. Vianello , Trigonometric Gaussian quadrature on subintervals of the period, Electronic Transactions on Numerical Analysis, 39 (2012) , 102-112. 
      M. Frank , Partial moment entropy approximation to radiative heat transfer, Pamm, 5 (2005) , 659-660.  doi: 10.1002/pamm.200510306.
      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.
      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.
      W. Fulton , Eigenvalues of sums of Hermitian matrices, Séminaire Bourbaki, 40 (1998) , 255-269. 
      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.
      E. M. Gelbard, Simplified Spherical Harmonics Equations and Their Use in Shielding Problems Technical Report WAPD-T-1182, Bettis Atomic Power Laboratory, 1961.
      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.
      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.
      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.
      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.
      M. Junk , Maximum entropy for reduced moment problems, Math. Meth. Mod. Appl. Sci., 10 (2000) , 1001-1025.  doi: 10.1142/S0218202500000513.
      C. Kelley, Solving Nonlinear Equations with Newton's Method Society for Industrial and Applied Mathematics, 2003. doi: 10.1137/1. 9780898718898.
      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.
      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.
      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.
      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.
      C. D. Levermore , Moment closure hierarchies for kinetic theories, Journal of Statistical Physics, 83 (1996) , 1021-1065.  doi: 10.1007/BF02179552.
      W. R. Martin, The application of the finite element method to the neutron transport equation, 1–232.
      G. N. Minerbo , Maximum entropy Eddington factors, J. Quant. Spectrosc. Radiat. Transfer, 20 (1978) , 541-545.  doi: 10.1016/0022-4073(78)90024-9.
      P. Monreal, Moment Realizability and Kershaw Closures in Radiative Transfer PhD thesis, TU Aachen, 2012.
      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.
      G. C. Pomraning , The Fokker-Planck operator as an asymptotic limit, Math. Mod. Meth. Appl. Sci., 2 (1992) , 21-36.  doi: 10.1142/S021820259200003X.
      A. Roth, Numerical Schemes for Kinetic Equations with Applications to Fibre Lay-Down and Interacting Particles Verlag Dr. Hut, 2014.
      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.
      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.
      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.
      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.
      F. Schneider, Moment Models in Radiation Transport Equations Dr. Hut Verlag, 2016.
      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.
      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.
      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.
      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.
  • 加载中

Figures(16)

Tables(3)

SHARE

Article Metrics

HTML views(185) PDF downloads(83) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return