Realizability-preserving DG-IMEX method for the two-moment model of fermion transport

Abstract

Building on the framework of Zhang & Shu [1,2], we develop a realizability-preserving method to simulate the transport of particles (fermions) through a background material using a two-moment model that evolves the angular moments of a phase space distribution function f . The two-moment model is closed using algebraic moment closures; e.g., as proposed by Cernohorsky & Bludman [3] and Banach & Larecki [4]. Variations of this model have recently been used to simulate neutrino transport in nuclear astrophysics applications, including core-collapse supernovae and compact binary mergers. We employ the discontinuous Galerkin (DG) method for spatial discretization (in part to capture the asymptotic diffusion limit of the model) combined with implicit-explicit (IMEX) time integration to stably bypass short timescales induced by frequent interactions between particles and the background. Appropriate care is taken to ensure the method preserves strict algebraic bounds on the evolved moments (particle density and flux) as dictated by Pauli's exclusion principle, which demands a bounded distribution function (i.e., f ∈ [0, 1]). This realizability-preserving scheme combines a suitable CFL condition, a realizability-enforcing limiter, a closure procedure based on Fermi-Dirac statistics, and an IMEX scheme whose stages can be written as a convex combination of forward Euler steps combined with a backward Euler step. The IMEX scheme is formally only first-order accurate, but works well in the diffusion limit, and -without interactions with the background -reduces to the optimal second-order strong stability-preserving explicit Runge-Kutta scheme of Shu & Osher [5]. Numerical results demonstrate the realizability-preserving properties of the scheme. We also demonstrate that the use of algebraic moment closures not based on Fermi-Dirac statistics can lead to unphysical moments in the context of fermion transport. (Ran Chu), endevee@ornl.gov (Eirik Endeve), hauckc@ornl.gov (Cory D. Hauck), mezz@utk.edu (Anthony Mezzacappa) both spectral and finite volume methods and are an attractive option for solving hyperbolic partial differential equations (PDEs). They achieve high-order accuracy on a compact stencil; i.e., data is only communicated with nearest neighbors, regardless of the formal order of accuracy, which can lead to a high computation to communication ratio, and favorable parallel scalability on heterogeneous architectures has been demonstrated [32]. Furthermore, they can easily be applied to problems involving curvilinear coordinates (e.g., beneficial in numerical relativity [33]). Importantly, DG methods exhibit favorable properties when collisions with a background are included, as they recover the correct asymptotic behavior in the diffusion limit, characterized by frequent collisions (e.g., [34,35,36]). The DG method was introduced in the 1970s by Reed & Hill [37] to solve the neutron transport equation, and has undergone remarkable developments since then (see, e.g., [38] and references therein).We are concerned with t...

Physical, numerical, and computational challenges of modeling neutrino transport in core-collapse supernovae

“…As such, it is an extension of the bound-preserving method discussed above, but for a nonlinear system of hyperbolic balance laws with stiff sources. As is reasonable for an initial investigation, the model adopted by Chu et al (2019) is rather simple, when compared to the two-moment models used to model neutrino transport in contemporary corecollapse supernova simulations. However, the work highlighted the role of the moment closure in the design of robust two-moment methods for neutrino transport, and developed an IMEX scheme with a reasonable time step restriction that is compatible with bounded solutions.…”

“…As such, the work put down the foundations for a framework that may help future developments of robust methods for models with improved physical fidelity. To simplify the discussion, we consider the model in Chu et al (2019) for one spatial dimension and define moments of the distribution function as where each P i ∈ V k h and each u i is a two-component vector representing the unknowns per element in the DG method. Then, for any x ∈ D and any ϕ h ∈ V k h , the semi-discrete DG method is as follows:…”

“…In the DG method, any standard numerical flux designed for hyperbolic conservation laws can be used. However, Chu et al (2019) used the global Lax-Friedrichs flux, where…”

“…To evolve Eq. (653) forward in time, Chu et al (2019) developed IMEX schemes, where the transport operator is treated explicitly and the collision operator is treated implicitly. As discussed in Sect.…”

“…This may be because the closures are very similar for the values of J and h encountered. Chu et al (2019) considered Eddington factors by Minerbo (1978), Cernohorsky and Bludman (1994), Larecki and Banach (2011) and Banach and Larecki (2017) and found that, under certain conditions, results obtained with closures based on Fermi-Dirac statistics can differ significantly from results obtained with the Minerbo (1978) closure, which is based on Boltzmann statistics.…”