A conservative orbital advection scheme for simulations of magnetized shear flows with the PLUTO code

Abstract

Context. Explicit numerical computations of hypersonic or super-fast differentially rotating disks are subject to the time-step constraint imposed by the Courant condition, according to which waves cannot travel more than a fraction of a cell during a single time-step update. When the bulk orbital velocity largely exceeds any other wave speed (e.g., sound or Alfvén), as computed in the rest frame, the time step is considerably reduced and an unusually large number of steps may be necessary to complete the computation. Aims. We present a robust numerical scheme to overcome the Courant limitation by improving and extending the algorithm previously known as FARGO (fast advection in rotating gaseous objects) to the equations of magnetohydrodynamics (MHD) using a more general formalism. The proposed scheme conserves total angular momentum and energy to machine precision and works in cartesian, cylindrical, or spherical coordinates. The algorithm has been implemented in the next release of the PLUTO code for astrophysical gasdynamics and is suitable for local or global simulations of accretion or proto-planetary disk models. Methods. By decomposing the total velocity into an average azimuthal contribution and a residual term, the algorithm approaches the solution of the MHD equations through two separate steps corresponding to a linear transport operator in the direction of orbital motion and a standard nonlinear solver applied to the MHD equations written in terms of the residual velocity. Since the former step is not subject to any stability restriction, the Courant condition is computed only in terms of the residual velocity, leading to substantially larger time steps. The magnetic field is advanced in time using the constrained transport method in order to fulfill the divergence-free condition. Furthermore, conservation of total energy and angular momentum is enforced at the discrete level by properly expressing the source terms in terms of upwind Godunov fluxes available during the standard solver. Results. Our results show that applications of the proposed orbital-advection scheme to problems of astrophysical relevance provides, at reduced numerical cost, equally accurate and less dissipative results than standard time-marching schemes.

Low-mass planets in nearly inviscid disks: numerical treatment

“…The first three codes in the list have been used and described in de Val-Borro et al (2006). The last code, PLUTO, is a multidimensional Riemann-solver based code for magnetohydrodynamical flows (Mignone et al 2007), which has been empowered recently with the FARGO-algorithm (Mignone et al 2012).…”

“…Meanwhile, similar orbital advection algorithms have been implemented into a variety of different codes in two and three spatial dimensions, e.g. NIRVANA (Ziegler & Yorke 1997;Kley et al 2009), ATHENA (Gardiner & Stone 2008;Stone & Gardiner 2010), and PLUTO (Mignone et al 2007(Mignone et al , 2012. Despite these widespread applications, it has been claimed recently that usage of the FARGO-algorithm (here in connection with ATHENA) may lead to an unsteady behavior of the flow near the planet, which even affects the wake structure of the flow farther away from the planet (Dong et al 2011b).…”

On the orbital evolution of a pair of giant planets in mean motion resonance

“…We note that for our simulations the options were chosen such that algorithm was applied using the residual azimuthal velocity with respect to the initial azimuthal velocity, the latter not being updated. A sample of runs were checked carefully to confirm numerical stability and that consistent results were obtained (see also Mignone et al 2012 for a comparison of this type). For the calculations reported here, we adopt a locally isothermal equation of state for which cs ∝ r −1/2 .…”

“…Mignone et al 2012, Uribe et al 2013. The planet positions are advanced using a fourth order Runge-Kutta method which however assumes that the forces due to the disc do not change as the planet locations are advanced through a time step, making the method one of lower order (see e.g.…”

Radiation magnetohydrodynamics in global simulations of protoplanetary discs

“…The PLUTO code is a highly modular, multidimensional, and multi-geometry code that can be applied to relativistic or non-relativistic (magneto-)hydrodynamics flows. For this work we chose the Godunov-type finite volume configuration which consists of a second order space reconstruction, a second-order Runge-Kutta time integration, the constrained transport (CT) method (Gardiner & Stone 2005), the orbital advection scheme FARGO MHD (Masset 2000;Mignone et al 2012), the HLLD Riemann solver (Miyoshi & Kusano 2005), and a Courant number of 0.3. In this work we neglect the magnetic dissipation which would appear at the right-hand side of Eq.…”