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

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., $fin[0,1]$. 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.

Conservative Moment Equations for Neutrino Radiation Transport with Limited Relativity

We derive conservative, multidimensional, energy-dependent moment equations for neutrino transport in core-collapse supernovae and related astrophysical systems, with particular attention to the consistency of conservative four-momentum and lepton number transport equations. After taking angular moments of conservative formulations of the general relativistic Boltzmann equation, we specialize to a conformally flat spacetime, which also serves as the basis for four further limits. Two of these---the multidimensional special relativistic case, and a conformally flat formulation of the spherically symmetric general relativistic case---are given in appendices for the sake of comparison with extant literature. The third limit is a weak-field, `pseudo-Newtonian approach citep{kim_etal_2009,kim_etal_2012} in which the source of the gravitational potential includes the trace of the stress-energy tensor (rather than just the mass density), and all orders in fluid velocity $v$ are retained. Our primary interest here is in the fourth limit: `$mathcal{O}(v)$ moment equations for use in conjunction with Newtonian self-gravitating hydrodynamics. We show that the concept of `$mathcal{O}(v)$ transport requires care when dealing with both conservative four-momentum and conservative lepton number transport, and present two self-consistent options: `$mathcal{O}(v)$-plus transport, in which an $mathcal{O}(v^2)$ energy equation combines with an $mathcal{O}(v)$ momentum equation to give an $mathcal{O}(v^2)$ number equation; and `$mathcal{O}(v)$-minus transport, in which an $mathcal{O}(v)$ energy equation combines with an $mathcal{O}(1)$ momentum equation to give an $mathcal{O}(v)$ number equation.

Turbulence and magnetic field amplification from spiral SASI modes in core-collapse supernovae

The stationary accretion shock instability (SASI) plays a central role in modern simulations of the explosion phase of core-collapse supernovae (CCSNe). It may be key to realizing neutrino powered explosions, and possibly links birth properties of pulsars (e.g., kick, spin, and magnetic field) to supernova dynamics. Using high-resolution magnetohydrodynamic simulations, we study the development of turbulence, and subsequent amplification of magnetic fields in a simplified model of the post-bounce core-collapse supernova environment. Turbulence develops from secondary instabilities induced by the SASI. Our simulations suggest that the development of turbulence plays an important role for the subsequent evolution of the SASI. The turbulence also acts to amplify weak magnetic fields via a small-scale dynamo.

Turbulent magnetic field amplification from spiral SASI modes in core-collapse supernovae

We describe the initial implementation of magnetohydrodynamics (MHD) in our astrophysical simulation code genasis. Then, we present MHD simulations exploring the capacity of the stationary accretion shock instability (SASI) to generate magnetic fields by adding a weak magnetic field to an initially spherically symmetric fluid configuration that models a stalled shock in the post-bounce supernova environment. Upon perturbation and nonlinear SASI development, shear flows associated with the spiral SASI mode contributes to a widespread and turbulent field amplification mechanism. While the SASI may contribute to neutron star magnetization, these simulations do not show qualitatively new features in the global evolution of the shock as a result of SASI-induced magnetic field amplification.

Generation of Magnetic Fields by the Stationary Accretion Shock Instability

We begin an exploration of the capacity of the stationary accretion shock instability (SASI) to generate magnetic fields by adding a weak, stationary, and radial (but bipolar) magnetic field, and in some cases rotation, to an initially spherically symmetric fluid configuration that models a stalled shock in the post-bounce supernova environment. In axisymmetric simulations we find that cycles of latitudinal flows into and radial flows out of the polar regions amplify the field parallel to the symmetry axis, typically increasing the total magnetic energy by about two orders of magnitude. Nonaxisymmetric calculations result in fundamentally different flows and a larger magnetic energy increase: shearing associated with the SASI spiral mode contributes to a widespread and turbulent field amplification mechanism, boosting the magnetic energy by almost four orders of magnitude (a result which remains very sensitive to the spatial resolution of the numerical simulations). While the SASI may contribute to neutron star magnetization, these simulations do not show qualitatively new features in the global evolution of the shock as a result of SASI-induced magnetic field amplification.

Magnetic field generation by the stationary accretion shock instability

By adding a weak magnetic field to a spherically symmetric fluid configuration that caricatures a stalled shock in the post-bounce supernova environment, we explore the capacity of the stationary accretion shock instability (SASI) to generate magnetic fields. The SASI develops upon perturbation of the initial condition, and the ensuing flow generates--{em in the absence of rotation}--dynamically significant magnetic fields ($sim 10^{15}$ G) on a time scale that is relevant for the explosion mechanism of core-collapse supernovae. We describe our model, present some recent results, and discuss their potential relevance for supernova models.