Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates
Abstract
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.
- Authors:
-
- 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
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Oak Ridge Leadership Computing Facility (OLCF); UT-Battelle LLC/ORNL, Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC)
- OSTI Identifier:
- 1565298
- Alternate Identifier(s):
- OSTI ID: 1464579
- Grant/Contract Number:
- AC05-00OR22725
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Journal of Computational Physics
- Additional Journal Information:
- Journal Volume: 287; Journal Issue: C; Journal ID: ISSN 0021-9991
- Publisher:
- Elsevier
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING; computer science; physics; boltzmann equation; radiation transport; hyperbolic conservation laws; discontinuous Galerkin; maximum principle; high order accuracy
Citation Formats
Endeve, Eirik, Hauck, Cory D., Xing, Yulong, and Mezzacappa, Anthony. Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates. United States: N. p., 2015.
Web. doi:10.1016/j.jcp.2015.02.005.
Endeve, Eirik, Hauck, Cory D., Xing, Yulong, & Mezzacappa, Anthony. Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates. United States. https://doi.org/10.1016/j.jcp.2015.02.005
Endeve, Eirik, Hauck, Cory D., Xing, Yulong, and Mezzacappa, Anthony. Wed .
"Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates". United States. https://doi.org/10.1016/j.jcp.2015.02.005. https://www.osti.gov/servlets/purl/1565298.
@article{osti_1565298,
title = {Bound-preserving discontinuous Galerkin methods for conservative phase space advection in curvilinear coordinates},
author = {Endeve, Eirik and Hauck, Cory D. and Xing, Yulong and Mezzacappa, Anthony},
abstractNote = {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.},
doi = {10.1016/j.jcp.2015.02.005},
journal = {Journal of Computational Physics},
number = C,
volume = 287,
place = {United States},
year = {Wed Apr 15 00:00:00 EDT 2015},
month = {Wed Apr 15 00:00:00 EDT 2015}
}
Web of Science
Works referenced in this record:
Discontinuous Finite Element Transport Solutions in Thick Diffusive Problems
journal, March 2001
- Adams, Marvin L.
- Nuclear Science and Engineering, Vol. 137, Issue 3
The piecewise-parabolic method in curvilinear coordinates
journal, September 1993
- Blondin, John M.; Lufkin, Eric A.
- The Astrophysical Journal Supplement Series, Vol. 88
Colloquium : Perspectives on core-collapse supernova theory
journal, February 2013
- Burrows, Adam
- Reviews of Modern Physics, Vol. 85, Issue 1
Conservative general relativistic Boltzmann equation
journal, July 2013
- Cardall, Christian Y.; Endeve, Eirik; Mezzacappa, Anthony
- Physical Review D, Vol. 88, Issue 2
Conservative formulations of general relativistic kinetic theory
journal, July 2003
- Cardall, Christian Y.; Mezzacappa, Anthony
- Physical Review D, Vol. 68, Issue 2
Positivity-preserving Lagrangian scheme for multi-material compressible flow
journal, January 2014
- Cheng, Juan; Shu, Chi-Wang
- Journal of Computational Physics, Vol. 257
Positivity-preserving discontinuous Galerkin schemes for linear Vlasov-Boltzmann transport equations
journal, January 2012
- Cheng, Yingda; Gamba, Irene M.; Proft, Jennifer
- Mathematics of Computation, Vol. 81, Issue 277
Positivity-preserving DG and central DG methods for ideal MHD equations
journal, April 2013
- Cheng, Yue; Li, Fengyan; Qiu, Jianxian
- Journal of Computational Physics, Vol. 238
Runge-Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
journal, September 2001
- Cockburn, Bernardo; Shu, Chi-Wang
- Journal of Scientific Computing, Vol. 16, Issue 3, p. 173-261
Strong Stability-Preserving High-Order Time Discretization Methods
journal, January 2001
- Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, Eitan
- SIAM Review, Vol. 43, Issue 1
Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit
journal, January 2010
- Guermond, Jean-Luc; Kanschat, Guido
- SIAM Journal on Numerical Analysis, Vol. 48, Issue 1
Explosion Mechanisms of Core-Collapse Supernovae
journal, November 2012
- Janka, Hans-Thomas
- Annual Review of Nuclear and Particle Science, Vol. 62, Issue 1
Nodal discontinuous Galerkin methods on graphics processors
journal, November 2009
- Klöckner, A.; Warburton, T.; Bridge, J.
- Journal of Computational Physics, Vol. 228, Issue 21
Explosion mechanism, neutrino burst and gravitational wave in core-collapse supernovae
journal, March 2006
- Kotake, Kei; Sato, Katsuhiko; Takahashi, Keitaro
- Reports on Progress in Physics, Vol. 69, Issue 4
Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II
journal, July 1989
- larsen, Edward W.; Morel, J. E.
- Journal of Computational Physics, Vol. 83, Issue 1
Interplay of Neutrino Opacities in Core-Collapse Supernova Simulations
journal, November 2012
- Lentz, Eric J.; Mezzacappa, Anthony; Messer, O. E. Bronson
- The Astrophysical Journal, Vol. 760, Issue 1
A Finite Difference Representation of Neutrino Radiation Hydrodynamics in Spherically Symmetric General Relativistic Spacetime
journal, January 2004
- Liebendorfer, Matthias; Messer, O. E. Bronson; Mezzacappa, Anthony
- The Astrophysical Journal Supplement Series, Vol. 150, Issue 1
Relativistic transport theory
journal, May 1966
- Lindquist, Richard W.
- Annals of Physics, Vol. 37, Issue 3
Discontinuous Galerkin finite element method applied to the 1-D spherical neutron transport equation
journal, April 2007
- Machorro, Eric
- Journal of Computational Physics, Vol. 223, Issue 1
Robust and accurate filtered spherical harmonics expansions for radiative transfer
journal, August 2010
- McClarren, Ryan G.; Hauck, Cory D.
- Journal of Computational Physics, Vol. 229, Issue 16
ASCERTAINING THE CORE COLLAPSE SUPERNOVA MECHANISM: The State of the Art and the Road Ahead
journal, December 2005
- Mezzacappa, Anthony
- Annual Review of Nuclear and Particle Science, Vol. 55, Issue 1
Computer simulation of time-dependent, spherically symmetric spacetimes containing radiating fluids - Formalism and code tests
journal, August 1989
- Mezzacappa, Anthony; Matzner, Richard A.
- The Astrophysical Journal, Vol. 343
On laboratory-frame radiation hydrodynamics
journal, October 2001
- Mihalas, Dimitri; Auer, L. H.
- Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 71, Issue 1
On the Solution of the Time-Dependent inertial-Frame Equation of radiative Transfer in Moving Media to
journal, August 1982
- Mihalas, Dimitri; Klein, Richard I.
- Journal of Computational Physics, Vol. 46, Issue 1
Three-Dimensional Boltzmann Hydro code for core Collapse in Massive Stars. i. Special Relativistic Treatments
journal, September 2014
- Nagakura, Hiroki; Sumiyoshi, Kohsuke; Yamada, Shoichi
- The Astrophysical Journal Supplement Series, Vol. 214, Issue 2
A realizability-preserving discontinuous Galerkin method for the M1 model of radiative transfer
journal, July 2012
- Olbrant, Edgar; Hauck, Cory D.; Frank, Martin
- Journal of Computational Physics, Vol. 231, Issue 17
Hyperbolic character of the angular moment equations of radiative transfer and numerical methods
journal, September 2000
- Pons, J. A.; Ibanez, J. M.; Miralles, J. A.
- Monthly Notices of the Royal Astronomical Society, Vol. 317, Issue 3
Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov–Poisson system
journal, September 2011
- Qiu, Jing-Mei; Shu, Chi-Wang
- Journal of Computational Physics, Vol. 230, Issue 23
A new spherical harmonics scheme for multi-dimensional radiation transport I. Static matter configurations
journal, June 2013
- Radice, David; Abdikamalov, Ernazar; Rezzolla, Luciano
- Journal of Computational Physics, Vol. 242
A general Eulerian formulation of the comoving-frame equation of radiative transfer
journal, November 1986
- Riffert, H.
- The Astrophysical Journal, Vol. 310
A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations
journal, July 2011
- Rossmanith, James A.; Seal, David C.
- Journal of Computational Physics, Vol. 230, Issue 16
Conservative form of Boltzmann’s equation in general relativity
journal, April 2014
- Shibata, Masaru; Nagakura, Hiroki; Sekiguchi, Yuichiro
- Physical Review D, Vol. 89, Issue 8
Relativistic radiative transfer: moment formalisms
journal, February 1981
- Thorne, K. S.
- Monthly Notices of the Royal Astronomical Society, Vol. 194, Issue 2
Positivity-preserving high order well-balanced discontinuous Galerkin methods for the shallow water equations
journal, December 2010
- Xing, Yulong; Zhang, Xiangxiong; Shu, Chi-Wang
- Advances in Water Resources, Vol. 33, Issue 12
High order positivity-preserving finite volume WENO schemes for a hierarchical size-structured population model
journal, October 2011
- Zhang, Rui; Zhang, Mengping; Shu, Chi-Wang
- Journal of Computational and Applied Mathematics, Vol. 236, Issue 5
On maximum-principle-satisfying high order schemes for scalar conservation laws
journal, May 2010
- Zhang, Xiangxiong; Shu, Chi-Wang
- Journal of Computational Physics, Vol. 229, Issue 9
Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments
journal, May 2011
- Zhang, Xiangxiong; Shu, Chi-Wang
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 467, Issue 2134
Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms
journal, February 2011
- Zhang, Xiangxiong; Shu, Chi-Wang
- Journal of Computational Physics, Vol. 230, Issue 4
Maximum-principle-satisfying second order discontinuous Galerkin schemes for convection–diffusion equations on triangular meshes
journal, February 2013
- Zhang, Yifan; Zhang, Xiangxiong; Shu, Chi-Wang
- Journal of Computational Physics, Vol. 234
Works referencing / citing this record:
Design of provably physical-constraint-preserving methods for general relativistic hydrodynamics
journal, May 2017
- Wu, Kailiang
- Physical Review D, Vol. 95, Issue 10
hp-adaptive discontinuous Galerkin solver for elliptic equations in numerical relativity
journal, October 2019
- Vincent, Trevor; Pfeiffer, Harald P.; Fischer, Nils L.
- Physical Review D, Vol. 100, Issue 8
Formulation of discontinuous Galerkin methods for relativistic astrophysics
text, January 2015
- Teukolsky, Saul A.
- arXiv
SpECTRE: A Task-based Discontinuous Galerkin Code for Relativistic Astrophysics
text, January 2016
- Kidder, Lawrence E.; Field, Scott E.; Foucart, Francois
- arXiv
Positivity preserving DG schemes for a Boltzmann - Poisson model of electrons in semiconductors in curvilinear momentum coordinates
preprint, January 2017
- Escalante, José A. Morales; Gamba, Irene M.; Endeve, Eirik
- arXiv
Entropy-stable positivity-preserving DG schemes for Boltzmann-Poisson models of collisional electronic transport along energy bands
preprint, January 2019
- Escalante, Jose A. Morales; Gamba, Irene M.
- arXiv