Bound-Preserving Discontinuous Galerkin Methods for Conservative Phase Space Advection in Curvilinear Coordinates

Technical Report · Sun Feb 01 04:00:00 EST 2015

DOI:https://doi.org/10.2172/1394128· OSTI ID:1394128

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

---

1. Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

We extend the positivity-preserving method of Zhang & Shu [49] 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 stabilitypreserving, 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 ϵ [0, 1]. The combination of suitable CFL conditions and the use of the high-order limiter proposed in [49] is su cient 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 divergencefree property of the phase space ow. 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.

• Research Organization: Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
• Sponsoring Organization: USDOE
• DOE Contract Number: AC05-00OR22725
• OSTI ID: 1394128
• Report Number(s): ORNL/TM-2014/571
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
boltzmann equation
discontinuous Galerkin
high order accuracy
hyperbolic conservation laws
maximum principle
radiation transport