High-order reconstruction schemes for the solution of hyperbolic conservation laws in orthogonal curvilinear coordinates are revised in the finite volume approach. The formulation employs a piecewise ...polynomial approximation to the zone-average values to reconstruct left and right interface states from within a computational zone to arbitrary order of accuracy by inverting a Vandermonde-like linear system of equations with spatially varying coefficients. The approach is general and can be used on uniform and non-uniform meshes although explicit expressions are derived for polynomials from second to fifth degree in cylindrical and spherical geometries with uniform grid spacing. It is shown that, in regions of large curvature, the resulting expressions differ considerably from their Cartesian counterparts and that the lack of such corrections can severely degrade the accuracy of the solution close to the coordinate origin. Limiting techniques and monotonicity constraints are revised for conventional reconstruction schemes, namely, the piecewise linear method (PLM), third-order weighted essentially non-oscillatory (WENO) scheme and the piecewise parabolic method (PPM).
The performance of the improved reconstruction schemes is investigated in a number of selected numerical benchmarks involving the solution of both scalar and systems of nonlinear equations (such as the equations of gas dynamics and magnetohydrodynamics) in cylindrical and spherical geometries in one and two dimensions. Results confirm that the proposed approach yields considerably smaller errors, higher convergence rates and it avoid spurious numerical effects at a symmetry axis.
•With this work we assess the validity of novel UCT schemes for the solution of the MHD equations in multiple dimensions.•The proposed schemes improve on the original formulation of Londrillo & Del ...Zanna (2004) 22 by providing more general guidelines allowing different Riemann solvers to be incorporated.•This approach offers enhanced flexibility allowing new upwind techniques to be incorporated in CT-MHD schemes at the modest cost of storing transverse velocity, weight coefficients for the fluxes and diffusion terms for the magnetic field.
The constrained transport (CT) method reflects the state of the art numerical technique for preserving the divergence-free condition of magnetic field to machine accuracy in multi-dimensional MHD simulations performed with Godunov-type, or upwind, conservative codes. The evolution of the different magnetic field components, located at zone interfaces using a staggered representation, is achieved by calculating the electric field components at cell edges, in a way that has to be consistent with the Riemann solver used for the update of cell-centered fluid quantities at interfaces. Albeit several approaches have been undertaken, the purpose of this work is, on the one hand, to compare existing methods in terms of robustness and accuracy and, on the other, to extend the upwind constrained transport (UCT) method by Londrillo & Del Zanna (2004) 22 and Del Zanna et al. (2007) 23 for the systematic construction of new averaging schemes. In particular, we propose a general formula for the upwind fluxes of the induction equation which simply involves the information available from the base Riemann solver employed for the fluid part, provided it does not require full spectral decomposition, and 1D reconstructions of velocity and magnetic field components from nearby intercell faces to cell edges. Our results are presented here in the context of second-order schemes for classical MHD, but they can be easily generalized to higher than second order schemes, either based on finite volumes or finite differences, and to other physical systems retaining the same structure of the equations, such as that of relativistic or general relativistic MHD.
ABSTRACT
We compare a particular selection of approximate solutions of the Riemann problem in the context of ideal relativistic magnetohydrodynamics. In particular, we focus on Riemann solvers not ...requiring a full eigenvector structure. Such solvers recover the solution of the Riemann problem by solving a simplified or reduced set of jump conditions, whose level of complexity depends on the intermediate modes that are included. Five different approaches – namely the HLL, HLLC, HLLD, HLLEM, and GFORCE schemes – are compared in terms of accuracy and robustness against one – and multidimensional standard numerical benchmarks. Our results demonstrate that – for weak or moderate magnetizations – the HLLD Riemann solver yields the most accurate results, followed by HLLC solver(s). The GFORCE approach provides a valid alternative to the HLL solver being less dissipative and equally robust for strongly magnetized environments. Finally, our tests show that the HLLEM Riemann solver is not cost-effective in improving the accuracy of the solution and reducing the numerical dissipation.
We describe an implementation of a particle physics module available for the PLUTO code appropriate for the dynamical evolution of a plasma consisting of a thermal fluid and a nonthermal component ...represented by relativistic charged particles or cosmic rays (CRs). While the fluid is approached using standard numerical schemes for magnetohydrodynamics, CR particles are treated kinetically using conventional Particle-In-Cell (PIC) techniques. The module can be used either to describe test-particle motion in the fluid electromagnetic field or to solve the fully coupled magnetohydrodynamics (MHD)-PIC system of equations with particle backreaction on the fluid as originally introduced by Bai et al. Particle backreaction on the fluid is included in the form of momentum-energy feedback and by introducing the CR-induced Hall term in Ohm's law. The hybrid MHD-PIC module can be employed to study CR kinetic effects on scales larger than the (ion) skin depth provided that the Larmor gyration scale is properly resolved. When applicable, this formulation avoids resolving microscopic scales, offering substantial computational savings with respect to PIC simulations. We present a fully conservative formulation that is second-order accurate in time and space, and extends to either the Runge-Kutta (RK) or the corner transport upwind time-stepping schemes (for the fluid), while a standard Boris integrator is employed for the particles. For highly energetic relativistic CRs and in order to overcome the time-step restriction, a novel subcycling strategy that retains second-order accuracy in time is presented. Numerical benchmarks and applications including Bell instability, diffusive shock acceleration, and test-particle acceleration in reconnecting layers are discussed.
We present a description of the adaptive mesh refinement (AMR) implementation of the PLUTO code for solving the equations of classical and special relativistic magnetohydrodynamics (MHD and RMHD). ...The current release exploits, in addition to the static grid version of the code, the distributed infrastructure of the CHOMBO library for multidimensional parallel computations over block-structured, adaptively refined grids. We employ a conservative finite-volume approach where primary flow quantities are discretized at the cell center in a dimensionally unsplit fashion using the Corner Transport Upwind method. Time stepping relies on a characteristic tracing step where piecewise parabolic method, weighted essentially non-oscillatory, or slope-limited linear interpolation schemes can be handily adopted. A characteristic decomposition-free version of the scheme is also illustrated. The solenoidal condition of the magnetic field is enforced by augmenting the equations with a generalized Lagrange multiplier providing propagation and damping of divergence errors through a mixed hyperbolic/parabolic explicit cleaning step. Among the novel features, we describe an extension of the scheme to include non-ideal dissipative processes, such as viscosity, resistivity, and anisotropic thermal conduction without operator splitting. Finally, we illustrate an efficient treatment of point-local, potentially stiff source terms over hierarchical nested grids by taking advantage of the adaptivity in time. Several multidimensional benchmarks and applications to problems of astrophysical relevance assess the potentiality of the AMR version of PLUTO in resolving flow features separated by large spatial and temporal disparities.
A new approximate Riemann solver for the equations of magnetohydrodynamics (MHD) with an isothermal equation of state is presented. The proposed method of solution draws on the recent work of Miyoshi ...and Kusano, in the context of adiabatic MHD, where an approximate solution to the Riemann problem is sought in terms of an average constant velocity and total pressure across the Riemann fan. This allows the formation of four intermediate states enclosed by two outermost fast discontinuities and separated by two rotational waves and an entropy mode. In the present work, a corresponding derivation for the isothermal MHD equations is presented. It is found that the absence of the entropy mode leads to a different formulation which is based on a three-state representation rather than four. Numerical tests in one and two dimensions demonstrate that the new solver is robust and comparable in accuracy to the more expensive linearized solver of Roe, although considerably faster.
ABSTRACT
High-resolution radio observations of cluster radio relics often show complex spatial and spectral features. However, it is not clear what these features reveal about the underlying magnetic ...field properties. We performed three-dimensional magnetohydrodynamical simulations of merger shock waves propagating through a magnetized, turbulent intracluster medium. Our model includes the diffusive shock acceleration (DSA) of cosmic ray electrons, their spatial advection and energy losses at run-time. With this set-up we can investigate the relation between radio substructure and pre-shock plasma conditions in the host cluster. We find that upstream turbulence plays a major role in shaping the properties of radio relics produced downstream. Within the assumption of DSA, we can reproduce the observed discrepancy between the X-ray derived Mach number of shocks, and the Mach number inferred from radio spectra. Our simulated spectral index maps and profiles across the radio relic also suggest that the standard deviation of the upstream magnetic field must be relatively small ($\sigma _B\le 1 \, \mu$G) in order to reproduce observations and therefore radio relics can potentially constrain the distribution of magnetic fields in galaxy clusters outskirts.
ABSTRACT
We carry out a linear stability analysis of a magnetized relativistic rotating cylindrical jet flow using the approximation of zero thermal pressure. We identify several modes of instability ...in the jet: Kelvin–Helmholtz, current-driven and two kinds of centrifugal-buoyancy modes – toroidal and poloidal. The Kelvin–Helmholtz mode is found at low magnetization and its growth rate depends very weakly on the pitch parameter of the background magnetic field and on rotation. The current-driven mode is found at high magnetization, the values of its growth rate and the wavenumber, corresponding to the maximum growth, increase as we decrease the pitch parameter of the background magnetic field. This mode is stabilized by rotation, especially, at high magnetization. The centrifugal-buoyancy modes, arising due to rotation, tend also to be more stable when magnetization is increased. Overall, relativistic jet flows appear to be more stable with respect to their non-relativistic counterpart.
Context. Extragalactic radio sources have been classified into two classes, Fanaroff-Riley I and II, which differ in morphology and radio power. Strongly emitting sources belong to the ...edge-brightened FR II class, and weakly emitting sources to the edge-darkened FR I class. The origin of this dichotomy is not yet fully understood. Numerical simulations are successful in generating FR II morphologies, but they fail to reproduce the diffuse structure of FR Is. Aims. By means of hydro-dynamical 3D simulations of supersonic jets, we investigate how the displayed morphologies depend on the jet parameters. Bow shocks and Mach disks at the jet head, which are probably responsible for the hot spots in the FR II sources, disappear for a jet kinetic power L sub(kin)< or = 10 super(43) ergs super(-1). This threshold compares favorably with the luminosity at which the FR I/FR II transition is observed. Methods. The problem is addressed by numerical means carrying out 3D HD simulations of supersonic jets that propagate in a non-homogeneous medium with the ambient temperature that increases with distance from the jet origin, which maintains constant pressure. Results. The jet energy in the lower power sources, instead of being deposited at the terminal shock, is gradually dissipated by the turbulence. The jets spread out while propagating, and they smoothly decelerate while mixing with the ambient medium and produce the plumes characteristic of FR I objects. Conclusions. Three-dimensionality is an essential ingredient to explore the FR I evolution becausethe properties of turbulence in two and three dimensions are very different, since there is no energy cascade to small scales in two dimensions, and two-dimensional simulations with the same parameters lead to FRII-like behavior.
ABSTRACT
We analyse the evolution of current-driven kink instabilities of a highly magnetized relativistic plasma column, focusing in particular on its dissipation properties. The instability ...evolution leads to the formation of thin current sheets where the magnetic energy is dissipated. We find that the total amount of dissipated magnetic energy is independent of the dissipation properties. Dissipation occurs in two stages: a peak when the instability saturates that is characterized by the formation of a helicoidal current sheet at the boundary of the deformed plasma column, followed by a weaker almost flat phase, in which turbulence develops. The detailed properties of these two phases depend on the equilibrium configuration and other parameters, in particular on the steepness of the pitch radial profile, on the presence of an external axial magnetic field and on the amount of magnetization. These results are relevant for high energy astrophysical sources, since current sheets can be the sites of magnetic reconnection where particles can be accelerated to relativistic energies and give rise to the observed radiation.
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