Pegasus: A New Hybrid-Kinetic Particle-in-Cell Code for Astrophysical Plasma Dynamics

We describe Pegasus, a new hybrid-kinetic particle-in-cell code tailored for the study of astrophysical plasma dynamics. The code incorporates an energy-conserving particle integrator into a stable, second-order--accurate, three-stage predictor-predictor-corrector integration algorithm. The constrained transport method is used to enforce the divergence-free constraint on the magnetic field. A delta-f scheme is included to facilitate a reduced-noise study of systems in which only small departures from an initial distribution function are anticipated. The effects of rotation and shear are implemented through the shearing-sheet formalism with orbital advection. These algorithms are embedded within an architecture similar to that used in the popular astrophysical magnetohydrodynamics code Athena, one that is modular, well-documented, easy to use, and efficiently parallelized for use on thousands of processors. We present a series of tests in one, two, and three spatial dimensions that demonstrate the fidelity and versatility of the code.Comment: 27 pages, 12 figures, accepted for publication in Journal of Computational Physic

Very High Order \PNM Schemes on Unstructured Meshes for the Resistive Relativistic MHD Equations

In this paper we propose the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, the one for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization we propose to use high order \PNM schemes as introduced in \cite{Dumbser2008} for hyperbolic conservation laws and a high order accurate unsplit time discretization is achieved using the element-local space-time discontinuous Galerkin approach proposed in \cite{DumbserEnauxToro} for one-dimensional balance laws with stiff source terms. The divergence free character of the magnetic field is accounted for through the divergence cleaning procedure of Dedner et al. \cite{Dedneretal}. To validate our high order method we first solve some numerical test cases for which exact analytical reference solutions are known and we also show numerical convergence studies in the stiff limit of the RRMHD equations using \PNM schemes from third to fifth order of accuracy in space and time. We also present some applications with shock waves such as a classical shock tube problem with different values for the conductivity as well as a relativistic MHD rotor problem and the relativistic equivalent of the Orszag-Tang vortex problem. We have verified that the proposed method can handle equally well the resistive regime and the stiff limit of ideal relativistic MHD. For these reasons it provides a powerful tool for relativistic astrophysical simulations involving the appearance of magnetic reconnection.Comment: 24 pages, 6 figures, submitted to JC

A Two-dimensional HLLC Riemann Solver for Conservation Laws : Application to Euler and MHD Flows

In this paper we present a genuinely two-dimensional HLLC Riemann solver. On logically rectangular meshes, it accepts four input states that come together at an edge and outputs the multi-dimensionally upwinded fluxes in both directions. This work builds on, and improves, our prior work on two-dimensional HLL Riemann solvers. The HLL Riemann solver presented here achieves its stabilization by introducing a constant state in the region of strong interaction, where four one-dimensional Riemann problems interact vigorously with one another. A robust version of the HLL Riemann solver is presented here along with a strategy for introducing sub-structure in the strongly-interacting state. Introducing sub-structure turns the two-dimensional HLL Riemann solver into a two-dimensional HLLC Riemann solver. The sub-structure that we introduce represents a contact discontinuity which can be oriented in any direction relative to the mesh. The Riemann solver presented here is general and can work with any system of conservation laws. We also present a second order accurate Godunov scheme that works in three dimensions and is entirely based on the present multidimensional HLLC Riemann solver technology. The methods presented are cost-competitive with traditional higher order Godunov schemes

A spherical shell numerical dynamo benchmark with pseudo vacuum magnetic boundary conditions

It is frequently considered that many planetary magnetic fields originate as a result of convection within planetary cores. Buoyancy forces responsible for driving the convection generate a fluid flow that is able to induce magnetic fields; numerous sophisticated computer codes are able to simulate the dynamic behaviour of such systems. This paper reports the results of a community activity aimed at comparing numerical results of several different types of computer codes that are capable of solving the equations of momentum transfer, magnetic field generation and heat transfer in the setting of a spherical shell, namely a sphere containing an inner core. The electrically conducting fluid is incompressible and rapidly rotating and the forcing of the flow is thermal convection under the Boussinesq approximation. We follow the original specifications and results reported in Harder & Hansen to construct a specific benchmark in which the boundaries of the fluid are taken to be impenetrable, non-slip and isothermal, with the added boundary condition for the magnetic field <b>B</b> that the field must be entirely radial there; this type of boundary condition for <b>B</b> is frequently referred to as ‘pseudo-vacuum’. This latter condition should be compared with the more frequently used insulating boundary condition. This benchmark is so-defined in order that computer codes based on local methods, such as finite element, finite volume or finite differences, can handle the boundary condition with ease. The defined benchmark, governed by specific choices of the Roberts, magnetic Rossby, Rayleigh and Ekman numbers, possesses a simple solution that is steady in an azimuthally drifting frame of reference, thus allowing easy comparison among results. Results from a variety of types of code are reported, including codes that are fully spectral (based on spherical harmonic expansions in angular coordinates and polynomial expansions in radius), mixed spectral and finite difference, finite volume, finite element and also a mixed Fourier-finite element code. There is good agreement among codes

Kinematic Dynamos using Constrained Transport with High Order Godunov Schemes and Adaptive Mesh Refinement

We propose to extend the well-known MUSCL-Hancock scheme for Euler equations to the induction equation modeling the magnetic field evolution in kinematic dynamo problems. The scheme is based on an integral form of the underlying conservation law which, in our formulation, results in a ``finite-surface'' scheme for the induction equation. This naturally leads to the well-known ``constrained transport'' method, with additional continuity requirement on the magnetic field representation. The second ingredient in the MUSCL scheme is the predictor step that ensures second order accuracy both in space and time. We explore specific constraints that the mathematical properties of the induction equations place on this predictor step, showing that three possible variants can be considered. We show that the most aggressive formulations (referred to as C-MUSCL and U-MUSCL) reach the same level of accuracy as the other one (referred to as Runge-Kutta), at a lower computational cost. More interestingly, these two schemes are compatible with the Adaptive Mesh Refinement (AMR) framework. It has been implemented in the AMR code RAMSES. It offers a novel and efficient implementation of a second order scheme for the induction equation. We have tested it by solving two kinematic dynamo problems in the low diffusion limit. The construction of this scheme for the induction equation constitutes a step towards solving the full MHD set of equations using an extension of our current methodology.Comment: 40 pages, 10 figures, accepted in Journal of Computational Physics. A version with full resolution is available at http://www.damtp.cam.ac.uk/user/fromang/publi/TFD.pd

A Hybrid Godunov Method for Radiation Hydrodynamics

From a mathematical perspective, radiation hydrodynamics can be thought of as a system of hyperbolic balance laws with dual multiscale behavior (multiscale behavior associated with the hyperbolic wave speeds as well as multiscale behavior associated with source term relaxation). With this outlook in mind, this paper presents a hybrid Godunov method for one-dimensional radiation hydrodynamics that is uniformly well behaved from the photon free streaming (hyperbolic) limit through the weak equilibrium diffusion (parabolic) limit and to the strong equilibrium diffusion (hyperbolic) limit. Moreover, one finds that the technique preserves certain asymptotic limits. The method incorporates a backward Euler upwinding scheme for the radiation energy density and flux as well as a modified Godunov scheme for the material density, momentum density, and energy density. The backward Euler upwinding scheme is first-order accurate and uses an implicit HLLE flux function to temporally advance the radiation components according to the material flow scale. The modified Godunov scheme is second-order accurate and directly couples stiff source term effects to the hyperbolic structure of the system of balance laws. This Godunov technique is composed of a predictor step that is based on Duhamel's principle and a corrector step that is based on Picard iteration. The Godunov scheme is explicit on the material flow scale but is unsplit and fully couples matter and radiation without invoking a diffusion-type approximation for radiation hydrodynamics. This technique derives from earlier work by Miniati & Colella 2007. Numerical tests demonstrate that the method is stable, robust, and accurate across various parameter regimes.Comment: accepted for publication in Journal of Computational Physics; 61 pages, 15 figures, 11 table

High-Resolution Mathematical and Numerical Analysis of Involution-Constrained PDEs

Partial differential equations constrained by involutions provide the highest fidelity mathematical models for a large number of complex physical systems of fundamental interest in critical scientific and technological disciplines. The applications described by these models include electromagnetics, continuum dynamics of solid media, and general relativity. This workshop brought together pure and applied mathematicians to discuss current research that cuts across these various disciplines’ boundaries. The presented material illuminated fundamental issues as well as evolving theoretical and algorithmic approaches for PDEs with involutions. The scope of the material covered was broad, and the discussions conducted during the workshop were lively and far-reaching