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Title: 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:
 [1];  [2];  [2];  [3]
  1. 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
  2. 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
  3. Univ. of Tennessee, Knoxville, TN (United States). Dept. of Physics and Astronomy
Publication Date:
Research Org.:
Oak Ridge National Lab. (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. https://doi.org/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 = {2015},
month = {4}
}

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