Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates

Endeve, Eirik; Hauck, Cory D.; Xing, Yulong; Mezzacappa, Anthony

doi:10.1016/j.jcp.2015.02.005

Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates

Journal Article · Wed Apr 15 00:00:00 EDT 2015 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2015.02.005· OSTI ID:1565298

Endeve, Eirik ^[1]; Hauck, Cory D. ^[2]; Xing, Yulong ^[2]; Mezzacappa, Anthony ^[3]

Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computational and Applied Mathematics Group; Univ. of Tennessee, Knoxville, TN (United States). Dept. of Physics and Astronomy
Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computational and Applied Mathematics Group; Univ. of Tennessee, Knoxville, TN (United States). Dept. of Mathematics
Univ. of Tennessee, Knoxville, TN (United States). Dept. of Physics and Astronomy

In this study, we extend the positivity-preserving method of Zhang & Shu to simulate the advection of neutral particles in phase space using curvilinear coordinates. The ability to utilize these coordinates is important for non-equilibrium transport problems in general relativity and also in science and engineering applications with specific geometries. The method achieves high-order accuracy using Discontinuous Galerkin (DG) discretization of phase space and strong stability-preserving, Runge-Kutta (SSP-RK) time integration. Special care in taken to ensure that the method preserves strict bounds for the phase space distribution function $$f$$; i.e., $$f\in[0,1]$$. The combination of suitable CFL conditions and the use of the high-order limiter proposed in Zhang & Shu (2010) is sufficient to ensure positivity of the distribution function. However, to ensure that the distribution function satisfies the upper bound, the discretization must, in addition, preserve the divergence-free property of the phase space flow. Proofs that highlight the necessary conditions are presented for general curvilinear coordinates, and the details of these conditions are worked out for some commonly used coordinate systems (i.e., spherical polar spatial coordinates in spherical symmetry and cylindrical spatial coordinates in axial symmetry, both with spherical momentum coordinates). Results from numerical experiments --- including one example in spherical symmetry adopting the Schwarzschild metric --- demonstrate that the method achieves high-order accuracy and that the distribution function satisfies the maximum principle.

View Accepted Manuscript (DOE)

Research Organization:: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF); UT-Battelle LLC/ORNL, Oak Ridge, TN (Unted States)

Sponsoring Organization:: USDOE Office of Science (SC)

Grant/Contract Number:: AC05-00OR22725

OSTI ID:: 1565298

Alternate ID(s):: OSTI ID: 1464579

Journal Information:: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 287; ISSN 0021-9991

Publisher:: ElsevierCopyright Statement

Country of Publication:: United States

Language:: English

References (40)

Discontinuous Galerkin methods Cockburn, B. ZAMM, Vol. 83, Issue 11 https://doi.org/10.1002/zamm.200310088	journal	November 2003
Relativistic transport theory Lindquist, Richard W. Annals of Physics, Vol. 37, Issue 3 https://doi.org/10.1016/0003-4916(66)90207-7	journal	May 1966
On the Solution of the Time-Dependent inertial-Frame Equation of radiative Transfer in Moving Media to Mihalas, Dimitri; Klein, Richard I. Journal of Computational Physics, Vol. 46, Issue 1 https://doi.org/10.1016/0021-9991(82)90007-9	journal	August 1982
Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II larsen, Edward W.; Morel, J. E. Journal of Computational Physics, Vol. 83, Issue 1 https://doi.org/10.1016/0021-9991(89)90229-5	journal	July 1989
On laboratory-frame radiation hydrodynamics Mihalas, Dimitri; Auer, L. H. Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 71, Issue 1 https://doi.org/10.1016/S0022-4073(01)00013-9	journal	October 2001
Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations Xing, Yulong; Zhang, Xiangxiong; Shu, Chi-Wang Advances in Water Resources, Vol. 33, Issue 12 https://doi.org/10.1016/j.advwatres.2010.08.005	journal	December 2010
High order positivity-preserving finite volume WENO schemes for a hierarchical size-structured population model Zhang, Rui; Zhang, Mengping; Shu, Chi-Wang Journal of Computational and Applied Mathematics, Vol. 236, Issue 5 https://doi.org/10.1016/j.cam.2011.05.007	journal	October 2011
Discontinuous Galerkin finite element method applied to the 1-D spherical neutron transport equation Machorro, Eric Journal of Computational Physics, Vol. 223, Issue 1 https://doi.org/10.1016/j.jcp.2006.08.020	journal	April 2007
Nodal discontinuous Galerkin methods on graphics processors Klöckner, A.; Warburton, T.; Bridge, J. Journal of Computational Physics, Vol. 228, Issue 21 https://doi.org/10.1016/j.jcp.2009.06.041	journal	November 2009
On maximum-principle-satisfying high order schemes for scalar conservation laws Zhang, Xiangxiong; Shu, Chi-Wang Journal of Computational Physics, Vol. 229, Issue 9 https://doi.org/10.1016/j.jcp.2009.12.030	journal	May 2010
Robust and accurate filtered spherical harmonics expansions for radiative transfer McClarren, Ryan G.; Hauck, Cory D. Journal of Computational Physics, Vol. 229, Issue 16 https://doi.org/10.1016/j.jcp.2010.03.043	journal	August 2010
Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms Zhang, Xiangxiong; Shu, Chi-Wang Journal of Computational Physics, Vol. 230, Issue 4 https://doi.org/10.1016/j.jcp.2010.10.036	journal	February 2011
A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations Rossmanith, James A.; Seal, David C. Journal of Computational Physics, Vol. 230, Issue 16 https://doi.org/10.1016/j.jcp.2011.04.018	journal	July 2011
Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system Qiu, Jing-Mei; Shu, Chi-Wang Journal of Computational Physics, Vol. 230, Issue 23 https://doi.org/10.1016/j.jcp.2011.07.018	journal	September 2011
A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer Olbrant, Edgar; Hauck, Cory D.; Frank, Martin Journal of Computational Physics, Vol. 231, Issue 17 https://doi.org/10.1016/j.jcp.2012.03.002	journal	July 2012
Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes Zhang, Yifan; Zhang, Xiangxiong; Shu, Chi-Wang Journal of Computational Physics, Vol. 234 https://doi.org/10.1016/j.jcp.2012.09.032	journal	February 2013
Positivity-preserving DG and central DG methods for ideal MHD equations Cheng, Yue; Li, Fengyan; Qiu, Jianxian Journal of Computational Physics, Vol. 238 https://doi.org/10.1016/j.jcp.2012.12.019	journal	April 2013
A new spherical harmonics scheme for multi-dimensional radiation transport I. Static matter configurations Radice, David; Abdikamalov, Ernazar; Rezzolla, Luciano Journal of Computational Physics, Vol. 242 https://doi.org/10.1016/j.jcp.2013.01.048	journal	June 2013
Positivity-preserving Lagrangian scheme for multi-material compressible flow Cheng, Juan; Shu, Chi-Wang Journal of Computational Physics, Vol. 257 https://doi.org/10.1016/j.jcp.2013.09.047	journal	January 2014
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems Cockburn, Bernardo; Shu, Chi-Wang Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261 https://doi.org/10.1023/A:1012873910884	journal	September 2001
Hyperbolic character of the angular moment equations of radiative transfer and numerical methods Pons, J. A.; Ibanez, J. M.; Miralles, J. A. Monthly Notices of the Royal Astronomical Society, Vol. 317, Issue 3 https://doi.org/10.1046/j.1365-8711.2000.03679.x	journal	September 2000
A general Eulerian formulation of the comoving-frame equation of radiative transfer Riffert, H. The Astrophysical Journal, Vol. 310 https://doi.org/10.1086/164726	journal	November 1986
Computer simulation of time-dependent, spherically symmetric spacetimes containing radiating fluids - Formalism and code tests Mezzacappa, Anthony; Matzner, Richard A. The Astrophysical Journal, Vol. 343 https://doi.org/10.1086/167756	journal	August 1989
The piecewise-parabolic method in curvilinear coordinates Blondin, John M.; Lufkin, Eric A. The Astrophysical Journal Supplement Series, Vol. 88 https://doi.org/10.1086/191834	journal	September 1993
A Finite Difference Representation of Neutrino Radiation Hydrodynamics in Spherically Symmetric General Relativistic Spacetime Liebendorfer, Matthias; Messer, O. E. Bronson; Mezzacappa, Anthony The Astrophysical Journal Supplement Series, Vol. 150, Issue 1 https://doi.org/10.1086/380191	journal	January 2004
Interplay of Neutrino Opacities in Core-Collapse Supernova Simulations Lentz, Eric J.; Mezzacappa, Anthony; Messer, O. E. Bronson The Astrophysical Journal, Vol. 760, Issue 1 https://doi.org/10.1088/0004-637X/760/1/94	journal	November 2012
Explosion mechanism, neutrino burst and gravitational wave in core-collapse supernovae Kotake, Kei; Sato, Katsuhiko; Takahashi, Keitaro Reports on Progress in Physics, Vol. 69, Issue 4 https://doi.org/10.1088/0034-4885/69/4/R03	journal	March 2006
Three-Dimensional Boltzmann Hydro code for core Collapse in Massive Stars. i. Special Relativistic Treatments Nagakura, Hiroki; Sumiyoshi, Kohsuke; Yamada, Shoichi The Astrophysical Journal Supplement Series, Vol. 214, Issue 2 https://doi.org/10.1088/0067-0049/214/2/16	journal	September 2014
Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations Cheng, Yingda; Gamba, Irene M.; Proft, Jennifer Mathematics of Computation, Vol. 81, Issue 277 https://doi.org/10.1090/S0025-5718-2011-02504-4	journal	January 2012
Relativistic radiative transfer: moment formalisms Thorne, K. S. Monthly Notices of the Royal Astronomical Society, Vol. 194, Issue 2 https://doi.org/10.1093/mnras/194.2.439	journal	February 1981
Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments Zhang, Xiangxiong; Shu, Chi-Wang Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 467, Issue 2134 https://doi.org/10.1098/rspa.2011.0153	journal	May 2011
Conservative formulations of general relativistic kinetic theory Cardall, Christian Y.; Mezzacappa, Anthony Physical Review D, Vol. 68, Issue 2 https://doi.org/10.1103/PhysRevD.68.023006	journal	July 2003
Conservative 3 + 1 general relativistic Boltzmann equation Cardall, Christian Y.; Endeve, Eirik; Mezzacappa, Anthony Physical Review D, Vol. 88, Issue 2 https://doi.org/10.1103/PhysRevD.88.023011	journal	July 2013
Conservative form of Boltzmann’s equation in general relativity Shibata, Masaru; Nagakura, Hiroki; Sekiguchi, Yuichiro Physical Review D, Vol. 89, Issue 8 https://doi.org/10.1103/PhysRevD.89.084073	journal	April 2014
Colloquium : Perspectives on core-collapse supernova theory Burrows, Adam Reviews of Modern Physics, Vol. 85, Issue 1 https://doi.org/10.1103/RevModPhys.85.245	journal	February 2013
Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit Guermond, Jean-Luc; Kanschat, Guido SIAM Journal on Numerical Analysis, Vol. 48, Issue 1 https://doi.org/10.1137/090746938	journal	January 2010
Strong Stability-Preserving High-Order Time Discretization Methods Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan SIAM Review, Vol. 43, Issue 1 https://doi.org/10.1137/S003614450036757X	journal	January 2001
Explosion Mechanisms of Core-Collapse Supernovae Janka, Hans-Thomas Annual Review of Nuclear and Particle Science, Vol. 62, Issue 1 https://doi.org/10.1146/annurev-nucl-102711-094901	journal	November 2012
ASCERTAINING THE CORE COLLAPSE SUPERNOVA MECHANISM: The State of the Art and the Road Ahead Mezzacappa, Anthony Annual Review of Nuclear and Particle Science, Vol. 55, Issue 1 https://doi.org/10.1146/annurev.nucl.55.090704.151608	journal	December 2005
Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems Adams, Marvin L. Nuclear Science and Engineering, Vol. 137, Issue 3 https://doi.org/10.13182/NSE00-41	journal	March 2001

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