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

TL;DR

This paper develops a high-order, bound-preserving discontinuous Galerkin method for simulating phase space advection in curvilinear coordinates, ensuring positivity and maximum principle adherence in complex geometries.

Contribution

It extends the positivity-preserving DG method to curvilinear coordinates with divergence-free flow conditions, enabling accurate, stable simulations in relativistic and geometrically complex settings.

Findings

Achieves high-order accuracy in phase space advection.

Ensures the distribution function remains within physical bounds.

Demonstrates effectiveness in spherical symmetry with Schwarzschild metric.

Abstract

We extend the positivity-preserving method of Zhang & Shu (2010, JCP, 229, 3091-3120) 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.