Implementing a new recovery scheme for primitive variables in the general relativistic magnetohydrodynamic code Spritz

General relativistic magnetohydrodynamic (GRMHD) simulations represent a fundamental tool to probe various underlying mechanisms at play during binary neutron star (BNS) and neutron star (NS)—black hole (BH) mergers. Contemporary flux-conservative GRMHD codes numerically evolve a set of conservative equations based on “conserved” variables which then need to be converted back into the fundamental (“primitive”) variables. The corresponding conservative-to-primitive variable recovery procedure, based on root-finding algorithms, constitutes one of the core elements of such GRMHD codes. Recently, a new robust, accurate and efficient recovery scheme called RePrimAnd was introduced, which has demonstrated the ability to always converge to a unique solution. The scheme provides fine-grained error policies to handle invalid states caused by evolution errors, and also provides analytical bounds for the error of all primitive variables. In this work, we describe the technical aspects of implementing the RePrimAnd scheme into the GRMHD code spritz. To check our implementation as well as to assess the various features of the scheme, we perform a number of GRMHD tests in three dimensions. Our tests, which include critical cases such as a NS collapse to a BH as well as the early evolution ($\sim 50\mathrm{ms}$) of a Fishbone-Moncrief BH-accrection disk system, show that RePrimAnd is able to support magnetized, low density environments with magnetic-to-fluid pressure ratios as high as ${10}^4$, in situations where the previously used recovery scheme fails.

Figure 1

Meridional snapshots of the evolution of a magnetized nonrotating NS, where the RePrimAnd C2P scheme with ideal gas EOS is adopted. Black lines correspond to $A_{\varphi }$ isocontours and reveal the geometry of the poloidal component of the magnetic field. The red line is the isodensity contour corresponding to 1% of the initial maximum density, whereas orange and yellow lines are the isodensity contours corresponding 100, and 5 times the artificial atmosphere density, respectively. The color scale indicates the magnetic field strength.

Figure 2

Evolution of normalized maximum rest-mass density (top) and magnetic field strength (bottom) for low (LR), medium (MR), and high resolution (HR) simulations, comparing also between the Noble C2P (dashed lines) scheme and the RePrimAnd (RPA) scheme, the latter either in conjunction with the ideal gas EOS (solid lines) or the hybrid EOS (dotted lines).

Figure 3

Relative change in total baryonic mass for the different tests with Noble and RePrimAnd schemes.

Figure 4

Fourier transform of the maximum rest-mass density evolution normalized to the amplitude of first frequency peak, for both the RePrimAnd and Noble schemes. The dash-dotted vertical lines represent the normal mode frequencies computed independently via a perturbative code as reported in [50].

Figure 5

Results for the test case of a nonrotating NS endowed with an external dipolar magnetic field, evolved with the RePrimAnd C2P scheme and using a low artificial atmosphere density of $6.3\times {10}^5\mathrm{g}/{\mathrm{cm}}^3$. The black lines are isocontours of the vector potential component $A_{\varphi }$. Rest mass density contours in red, orange, and yellow are illustrated in the same fashion as done in Fig. 1.

Figure 6

Results for the external dipole test obtained with the Noble C2P scheme. Left: Same as the middle panel of Fig. 5, but showing the instability developing when adopting the Noble scheme. Right: Snapshot after 10 ms evolution with the Noble C2P scheme when using an artificial atmosphere density of $6.3\times {10}^7\mathrm{g}/{\mathrm{cm}}^3$, enlarged by a factor 100 compared to the RePrimAnd simulation results shown in the right panel of Fig. 5.

Figure 7

Evolution of normalized maximum rest-mass density (top), normalized magnetic field strength at the center (middle), and magnetic energy (bottom) for the case of a nonrotating NS endowed with extended dipolar magnetic field, evolved with the RePrimAnd C2P scheme and artificial atmosphere density of ${\rho }_{\mathrm{atm}}=6.3\times {10}^5\mathrm{g}/{\mathrm{cm}}^3$ or with the Noble C2P scheme and ${\rho }_{\mathrm{atm}}=6.3\times {10}^7\mathrm{g}/{\mathrm{cm}}^3$.

Figure 8

Meridional view of a magnetized rotating NS, evolved using the RePrimAnd C2P scheme. Color scale indicates the magnetic field strength and the isodensity contours are the same as in Fig. 1. Left panel shows the time at which an extended dipolar magnetic field is imposed (10 ms), while the right panel shows the system toward the end of the simulation.

Figure 9

Three dimensional magnetic field structure of the system shown in right panel of Fig. 8. A sphere of radius 10 km is added in the center to give a spatial scale reference.

Figure 10

Meridional snapshots of magnetic field strength for the magnetized rotating NS collapse simulation, evolved using the RePrimAnd C2P scheme. The isodensity contours in red, orange, and yellow correspond to 5, 3 and 1.5 times the artificial atmosphere density, respectively. The left panel, referring to the initial setup, shows also the magnetic field geometry as $A_{\varphi }$ isocontours (black). The middle and right panels refer to 1 and 1.5 ms of evolution, respectively. The $A_{\varphi }$ isocontours are not shown in the last two panels as their disordered structure overcrowds the regions of interest we want to show. The black area marks the apparent horizon of the BH.

Figure 11

Evolution of maximum rest-mass density normalized to the initial value (top left), total magnetic energy (bottom left), minimum of lapse (top right), and L2 norm of the Hamiltonian constraint (bottom right). Once the BH is formed, the interior of the apparent horizon is excluded for the computation of these quantities. The vertical black (dashed) line in all panels marks the time at which the apparent horizon is found in the simulation, while the horizontal yellow line in the top-left panel represents the floor density.

Figure 12

Initial and final time snapshots of the rest-mass density in the meridional (top row) and equatorial (bottom row) planes for the magnetized Fishbone-Moncrief BH-accretion disk test. The apparent horizon is depicted in black.

Figure 13

Initial and final time snapshots of the magnetic field strength within the meridional (top-row) and equatorial (bottom-row) planes, for the magnetized Fishbone-Moncrief BH-accretion disk test. The apparent horizon is depicted in black. In the lower left panel, we also highlight the mesh-refinement boundaries (red).

Figure 14

Time evolution of the mass accretion rate (top-panel) and total magnetic energy (bottom-panel) for the magnetized Fishbone-Moncrief BH-accretion disk test.

Figure 15

Shock profiles at time $t=0.4$ for test cases A and B from Table 1, evolved while using the RePrimAnd and Noble C2P schemes, in comparison to the exact solution computed using the code of [71].