Quantifying uncertainties in general relativistic magnetohydrodynamic codes Pedro L. Espino1 2Gabriele Bozzola3and Vasileios Paschalidis3 4 1Department of Physics The Pennsylvania State University University Park PA 16802 USA

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Quantifying uncertainties in general relativistic magnetohydrodynamic codes

Pedro L. Espino,1, 2 Gabriele Bozzola,3and Vasileios Paschalidis3, 4

1Department of Physics, The Pennsylvania State University, University Park, PA 16802, USA

2Department of Physics, University of California, Berkeley, CA 94720, USA

3Department of Astronomy, University of Arizona, Tucson, AZ 85721,USA

4Department of Physics, University of Arizona, Tucson, AZ 85721, USA

(Dated: October 27, 2022)

In this paper, we show that similar open-source codes for general relativistic magnetohydrody-

namic (GRMHD) produce diﬀerent results for key features of binary neutron star mergers. First,

we present a new open-source version of the publicly available IllinoisGRMHD code that provides

support for realistic, ﬁnite temperature equations of state. After stringent tests of our upgraded

code, we perform a code comparison between GRHydro,IllinoisGRMHD,Spritz, and WhiskyTHC,

which implement the same physics, but slightly diﬀerent computational methods. The beneﬁt of

the comparison is that all codes are embedded in the EinsteinToolkit suite, hence their only diﬀer-

ence is algorithmic. We ﬁnd similar convergence properties, ﬂuid dynamics, and gravitational waves,

but diﬀerent merger times, remnant lifetimes, and gravitational wave phases. Such diﬀerences must

be resolved before the post-merger dynamics modeled with such simulations can be reliably used to

infer the properties of nuclear matter especially in the era of precision gravitational wave astronomy.

I. INTRODUCTION

The detection of gravitational waves (GWs) from the

merger of a binary neutron star (BNS) [1] has established

the need for a clearer understanding of BNS systems. To

gain insight into the physics of BNS mergers, observa-

tions associated with such events – be they in the GW

or electromagnetic (EM) spectrum – must be analyzed

with either simpliﬁed analytical models which parame-

terize the dominant physical phenomena in the system,

e.g., [2–6] or with the use of accurate numerical relativity

(NR) simulations. Parametric/phenomenological mod-

els of BNS mergers cannot reliably capture the physics

of the most extreme stages of the merger, including the

merger itself and the environment immediately following,

where the matter is in a highly dynamical state and the

spacetime curvature is the strongest. For a reliable, ﬁrst-

principles understanding of these stages of BNS mergers,

the use of NR is the only recourse. Numerical simula-

tions allow for the systematic isolation of diﬀerent phys-

ical phenomena/physics, which provides a powerful tool

for deducing the role of magnetic ﬁelds, e.g., [7–14], equa-

tion of state (EOS) eﬀects, e.g., [15–24], and neutrino

transport, e.g., [18,20,25,26].

There are many important physical eﬀects to consider

in simulations of BNS mergers (see [27–32] for reviews);

such as ﬁnite temperature EOSs and magnetic ﬁelds. Re-

garding the use of realistic EOSs in BNS merger simula-

tions, an important aspect is the consistent evolution of

the electron fraction, as it provides crucial information

for the description of nucleosynthesis processes that take

part during and after merger, and is necessary to fully

describe kilonova (KN) afterglows [33,34]. Addition-

ally, the thermodynamic properties of the post-merger

remnant and ejecta are important and not well approxi-

mated by cold EOSs (see, e.g., [35,36]). Magnetic ﬁelds

are also a crucial ingredient to consider in BNS merger

simulations. In particular, an area which remains poorly

understood is the eﬀect of magnetic ﬁelds on ejecta prop-

erties. Large scale magnetic ﬁelds are important in the

evolution of the post-merger remnant and in the kilo-

nova signal associated with BNS mergers, as they are

expected to power relativistic jets [13,37–40] and drive

relativistic outﬂow [41–43]. The interplay between ﬁnite

temperature EOSs and strong magnetic ﬁelds remains

insuﬃciently characterized in BNS merger simulations.

For example, realistic EOSs have been shown to produce

lower velocity ejecta when compared to simple analytic

EOSs [18,23,31,44,45] while, on the other hand, sim-

ulations which account for magnetic eﬀects suggest that,

if magnetic ﬁelds are strong enough, shocked dynamical

ejecta during merger could be boosted to higher veloci-

ties [10–14,46]. Most modern simulations of BNS merg-

ers consider only some of the aforementioned physical ef-

fects, and there have only been a handful of simulations

which consider all eﬀects at once [10–14,47]. Even in

the case of more complete simulations, all treatments of

neutrino transport to-date must make some level of ap-

proximation since solving the full 6+1 dimensional neu-

trino transport equation at is not possible with current

schemes and computational resources.

There are currently available several codes with sup-

port to diﬀerent levels of physics. For instance, many

state-of-the-art codes with microphysical EOS compat-

ibility do not implement a constrained-transport treat-

ment of magnetic ﬁelds. While variety in the numerical

methods and codes used in the study of BNS mergers is

a strong-point from the perspective of understanding nu-

merical systematic errors, key diﬀerences between codes

can lead to a range of predictions for the relevant ob-

servables, depending on the physics included in simula-

tions. For example, although there is general agreement

in the ejecta properties predicted by diﬀerent codes [17–

19,23,25,45,48–50], dynamical ejecta masses range

over 10−4M.Mej .10−2Mwhile speeds range over

0.1c.vej .0.3c, depending on the total binary mass,

arXiv:2210.13481v2 [gr-qc] 26 Oct 2022

EOS, and numerical methods considered. Simulations

with even the highest resolutions can have numerical un-

certainties of over 40% [48,51–53]. The need for accurate

simulations and detailed estimates of the error budgets is

becoming more and more urgent as future GW detectors

are expected to be more sensitive than current system-

atic errors [54]. Future investigations of BNS mergers

would ideally require a set of catch-all, accurate, open

source GRMHD codes which can reliably simulate BNS

systems while considering as many microphysical phe-

nomena as possible. At the very least, reliable future

predictions using NR codes require detailed comparisons,

including cross-checks of the results obtained using dif-

ferent codes and of the varied numerical methods used

within the codes themselves.

In order to begin addressing each of these needs, we

ﬁrst present a new implementation of realistic, ﬁnite-

temperature EOS support in IllinoisGRMHD1(as ex-

tensively discussed below), and then perform a system-

atic comparison between diﬀerent open-source GRMHD

codes: IllinoisGRMHD [56], GRHydro [57], Spritz [58],

and WhiskyTHC [59–61]. These codes implement the same

physics and have largely overlapping numerical methods,

e.g., they are all housed within the EinsteinToolkit [62,

63], thereby we can isolate and compare the impact on

the evolution and uncover the existence of systematic er-

rors arising only from the GRMHD codes. This allows

us to test the robustness of certain results with respect

to diﬀerent computational choices. In most cases, such

choices are about failsafes for when the main algorithms

fail, as in the case of the conservative-to-primitive scheme

used in the artiﬁcial atmosphere. In our comparison, we

focused on BNS mergers and found that important quan-

tities disagree across the codes. For example, codes do

not agree on whether the remnant is stable or undergoes

collapse very shortly after merger, which has profound

implications for EM observations. This calls for more

detailed studies and comparisons across diﬀerent codes.

This paper is split in two parts. First, we discuss in de-

tail the formalism employed by IllinoisGRMHD, the ex-

tensions required to reach feature-parity with the other

codes, and the tests we performed. In particular, in

Sec. II we review the basic equations that are relevant

to BNS systems, and provide detail on their numerical

treatment within IllinoisGRMHD. In Sec. III we describe

our methods for implementing realistic, ﬁnite tempera-

ture EOS capability within IllinoisGRMHD. In Sec. IV

we discuss stringent dynamical tests of the extended ver-

sion of IllinoisGRMHD which cover a broad range of sce-

narios. Readers that are familiar with GRMHD simula-

tions may skip this part and focus on the results. In the

second part, starting from Section IV E, we discuss the

results from our simulations of BNS mergers.

Our extension to IllinoisGRMHD to allow for the use

of realistic, ﬁnite-temperature EOSs is public. The ex-

1Recently, [55] implemented a similar extension to IllinoisGRMHD.

tensions we have made to IllinoisGRMHD are crucial for

understanding the interplay between magnetic ﬁelds and

the EOS in the post-merger environment, and is a ﬁrst

step to including neutrino transport schemes in the code.

The current state of IllinoisGRMHD (including the ex-

tensions we describe here) opens up many avenues of

investigation surrounding compact object mergers with

strong magnetic ﬁelds. For instance, the new code capa-

bilities will make possible the calculation of nucleosyn-

thesis rates and help elucidate the role of microphysi-

cal, ﬁnite temperature EOSs in BNS mergers with strong

magnetic ﬁelds, among many other interesting phenom-

ena.

Throughout the work we use geometrized units, where

G=c= 1, unless otherwise stated. In addition, in

cases where we use logarithmic scales, we assume that

log ≡log10, unless otherwise noted. All visualizations

and post-process analyses in this work were carried out

with the kuibit software package [64].

II. BASIC EQUATIONS

Throughout this work we will predominantly work

with numerical codes that solve the Einstein ﬁeld equa-

tions,

Gµν = 8πTµν ,(1)

(where Gµν and Tµν are the Einstein and stress-

energy tensors, respectively), coupled to the equations

of ideal relativistic (magneto)hydrodynamics. In par-

ticular, we focus on the use of IllinoisGRMHD to solve

the equations of ideal relativistic magnetohydrodynam-

ics. IllinoisGRMHD takes advantage of the Baumgarte-

Shapiro-Shibata-Nakamura (BSSN) formulation [65–67]

of the 3+1 Arnowitt-Deser-Misner (ADM) formalism,

which recasts the Einstein equations in the form of an

initial-value problem (for more details, see textbooks on

the subject, e.g. [68–71]), in which the spacetime line el-

ement is given by

ds2=−α2dt2+γij (dxi+βidt)(dxj+βjdt),(2)

where αis the lapse, βiis the shift vector, γµν =

gµν +nµnνrepresents the induced metric on spacelike

hypersurfaces, and nµ= (1/α, −βi/α) is the future-

pointing unit vector orthogonal to each spatial slice. The

solution to Eq. (1) involves the evolution of the magne-

tohydrodynamic variables that appear on the right hand

sides of the ADM equations. An approach which is well

suited to the 3+1 formalism is the Eulerian (or Valencia)

formulation of relativistic hydrodynamics [72,73]. In this

formulation, the evolution equations for the relevant ﬂuid

variables arise from several conservation laws, including

the continuity equation

∇µ(ρbuµ) = 0,(3)

where ρbis the rest mass density and uµis the ﬂuid four-

velocity, the lepton number conservation (when neutrino

eﬀects are ignored)

uµ∇µ(Ye) = 0,(4)

which can be rewritten by use of Eq. (3) as

∇µ(ρbYeuµ) = 0,(5)

where Ye≡ne/nbis the electron fraction and nb(ne) is

the baryon (electron) number density, the conservation

of stress-energy

∇µTµν = 0,(6)

and the homogeneous Maxwell’s equations

∇νF∗µν = 0,(7)

where F∗µν =1

2εµναβ Fαβ is the dual to the electromag-

netic ﬁeld strength tensor Fαβ and εµναβ is the rank-4

Levi-Civita symbol. The matter variables are evolved

using Eqs. (3)-(7) once they have been cast in ﬂux-

conservative form,

∂tC+∇ · F=S,(8)

where C,F, and Sare vectors which contain the conser-

vative, ﬂux, and source terms, respectively. The vector

of primitive variables P

P=





ρb







,(9)

contains information on the physical state of the ﬂuid,

where Pis the ﬂuid pressure, vi=ui/u0is the ﬂuid

three-velocity and Biare the spatial components of the

magnetic ﬁeld as measured by a normal observer. The

conservative variables Care determined in terms of the

primitive variables, the lapse α, and the metric as

C=





ρ∗

˜τ







=





α√γρbu0

α2√γT 00 −ρ∗

(ρ∗h+αu0b2)ui−α√γb0bi

√γBi

ρ∗Ye







,(10)

where bµ=Bµ

(u)/√4π(where Bµ

(u)is the magnetic ﬁeld as

measured by an observer in the ﬂuid rest-frame), γis the

determinant of the 3-metric, and h= 1 + +P/ρ0is the

speciﬁc enthalpy (where is the speciﬁc internal energy).

We note that IllinoisGRMHD uses the coordinate three-

velocity, vi≡ui/u0, unlike many other evolution codes,

which also adopt the Valencia formalism, but use the Eu-

lerian three-velocity vi

(n)=ui/(αu0) + βi/α [74,75], i.e.,

the velocity measured by normal observers. Addition-

ally, we note that IllinoisGRMHD works with an ideal

ﬂuid stress-energy tensor of the form

Tµν = (ρbh+b2)uµuν+ (P+b2/2)gµν −bµbν.(11)

The evolution equations for the relevant ﬂuid variables

are determined by using the aforementioned conservation

laws and casting them in the form of Eq. (8). It is useful

to pay special attention to the evolution of the magnetic

ﬁeld, due to how it is treated in IllinoisGRMHD. Specif-

ically, the evolution equation for the magnetic ﬁeld is

∂t˜

Bi+∂j(vj˜

Bi−vi˜

Bj)=0,(12)

where ˜

Biis the conservative variable corresponding to

the magnetic ﬁeld Bi, as deﬁned in Eq. (10). The use

of Eq. (12) to evolve the magnetic ﬁeld results in terms

that violate the no-monopole constraint (∇ · B= 0),

which is addressed in IllinoisGRMHD by instead consid-

ering the evolution of the vector potential Aµ= Φnµ+Aµ

(where Aiis the magnetic vector potential and Φ is elec-

tric scalar potential) and recovering the magnetic ﬁeld as

Bi=εijk∂jAk. The evolution equation for Aiis

∂tAi=εijkvj˜

Bk−∂i(αΦ−βjAj).(13)

IllinoisGRMHD works in the generalized Lorenz gauge

∇µAµ=ξnµAµ[76], where ξis chosen such that the

Courant-Friedrich-Lewy (CFL) condition corresponding

to it is satisﬁed at all times given the grid choices.

In the following we provide additional descriptions of

the algorithms employed within IllinoisGRMHD for the

implementation of ﬁnite temperature EOS compatibil-

ity. We direct the reader to [56] for a detailed descrip-

tion of all of the additional algorithms employed within

IllinoisGRMHD.

III. IMPLEMENTATION OF REALISTIC

EQUATION OF STATE COMPATIBILITY

WITHIN ILLINOISGRMHD

The current open-source version of IllinoisGRMHD

solves the equations of GRMHD by assuming simple, an-

alytic EOSs, such as Γ-law or piecewise polytropic EOSs

with hybrid thermal eﬀects. These EOSs can only pro-

vide a qualitative understanding of the state of matter

during a BNS merger [24,35]. However, eﬀorts to model

parametrically both the thermal and the cold component

of the nuclear EOS in BNS mergers are under way, see,

e.g., [24,77–79]

The implementation of realistic EOSs within

IllinoisGRMHD allows us to understand, in a more

detailed manner, the interplay between the EOS,

thermal eﬀects, and magnetic ﬁelds in these systems.

Moreover, it is a crucial ﬁrst step toward imple-

menting additional important microphysics within

IllinoisGRMHD, such as neutrino transport. The inclu-

sion of realistic EOS capability within IllinoisGRMHD

required two steps: the implementation of the evolution

equation for the electron fraction and the implemen-

tation of algorithms which can perform the non-trivial

inversion of the evolved conservative variables Cto the

physical primitive variables P, which we refer to as

conservative-to-primitive inversion. In the following we

discuss the implementation of the evolution equation

for the electron fraction Yewithin our extended version

of IllinoisGRMHD. We also discuss the implementation

of state-of-the-art routines within IllinoisGRMHD for

conservative-to-primitive inversion and present relevant

tests. For the remainder of this work, we refer to

the currently available version of IllinoisGRMHD as

OriginalIllinoisGRMHD (abbreviated as OIL). Our

extended version of IllinoisGRMHD will be referred to

as MicrophysicalIllinoisGRMHD (abbreviated as MIL).

In cases where we discuss features which are common

between the two codes, we will refer to them jointly as

the IllinoisGRMHD code. Algorithmically, OIL and MIL

are identical, except for the changes highlighted in this

work which are required for realistic EOS compatibility.

A. Evolution of the electron fraction

In ﬂux-conservative form, the electron fraction evolu-

tion equation is

∂t(˜

Ye) + ∂j(vj˜

Ye)=0.(14)

With the inclusion of Eq. (14), the full set of equations

solved within MIL is

∂t





ρ∗

˜τ







+∂j





ρ∗vj

Yevj

α2√γT 0j−ρ∗vj

α√γT j

αΦ−βjAj







=





2α√γT αβ gαβ,i

ijkvj˜







(15)

where

s=α√γ[(T00βiβj+ 2T0iβj+Tij )Kij

−(T00βi+T0i)∂iα],(16)

and Kij is the extrinsic curvature. The evolution

of ˜

Yefollows that of the other conservative variables,

which begins with the determination of initial conditions.

Presently, we allow for Yeto be initialized in two possible

ways:

1. Linear proﬁle: we set Ye= Υρb(where Υ is a

constant that ensures proper dimensionality), such

that the electron fraction proﬁle is linear with re-

spect to ρb, in order to consider a simple proﬁle

where gradients are non-zero throughout the solu-

tion grid. This initialization is unphysical and only

useful for testing the advection of Yein situations

without realistic EOSs, i.e., passive advection of the

electron fraction.

2. β-equilibrium proﬁle: we set Ye(ρb) according to

the conditions for β-equilibrium in cold neutron

star (NS) matter. This initial proﬁle is suitable

for realistic descriptions of isolated stars as well

as for binaries that are initially separated at large

enough distances such that the components are

cold. All of the BNS initial data considered in

our tests are built for quasi-equilibrium systems,

in which the assumption that the components

are cold, β-equilibrated stars is well justiﬁed. We

use the same assumption of cold β-equilibrium

to initialize all other hydrodynamical variables at

t= 0.

Once the initial data are speciﬁed for the primitive vari-

ables at all grid points, the conservative variables are

obtained through the simple algebraic relations provided

in Eq. (10), which provides Cat all grid points. For

the evolution of C, we employ 3 ghost-zones at the outer

boundary of each adaptive mesh reﬁnement (AMR) grid.

We ﬁll all buﬀer zones at the reﬁnement level bound-

ary with data interpolated from neighboring rougher or

ﬁner levels of reﬁnement using standard prolongation or

restriction operators, respectively. Filling in the buﬀer

zones in this manner results in Cbeing either prolon-

gated or restricted after calculation [56]. To ensure con-

sistency between Cand P, the primitives are recovered

from the conservatives using a root-ﬁnding algorithm at

t= 0, which we discuss in Sec. III B. Next, we eval-

uate the ﬂux term Fin preparation for the next time

step. To this end, we must reconstruct the primitives be-

tween grid points (i.e, at cell interfaces). IllinoisGRMHD

employs the piecewise parabolic method (PPM) [80] for

primitives reconstruction. Reconstruction is used to eval-

uate the primitives on the left and right interfaces of all

grid points, PL,R, in all directions. These interface values

are then used to calculate the corresponding conservative

variables at cell interfaces, which are in turn used to cal-

culate the ﬂux term in Eq. (14) using a second-order,

ﬁnite-volume, high-resolution shock-capturing (HRSC)

scheme. The handling of ﬂuxes at grid interfaces FL,R re-

quires a solution to a Riemann problem. IllinoisGRMHD

employs the standard Harten-Lax-vanLeer (HLL) [81] ap-

proximate Riemann solver, where for a given direction

the electron fraction ﬂux is given by

FHLL(Ye) = c−FR+c+FL−c+c−(˜

Ye,R −˜

Ye,L)

c++c−,(17)

where c±=±max 0, cR

±, cL

±and cL,R

±are the maximum

(+) and minimum (−) characteristic speeds at the left

(L) and right (R) cell interfaces (see [56,82] for further

algorithmic details). The derivatives of the ﬂuxes are

then determined and summed independently for each di-

rection. For instance, the ﬂux along the x-direction takes

the form

(∂xFx)ijk =

FHLL,x

i+1

2jk (Ye)−FHLL,x

i−1

2jk (Ye)

∆x.(18)

The ﬂuxes along the y- and z-directions take a similar

form, but we instead consider ﬁnite diﬀerencing along

the jand kindices, respectively.

The evolution equation for Yedoes not include source

terms in the absence of neutrinos, so the right-hand-side

of Eq. (18) is then passed to the method of lines (MoL)

thorn, which integrates the conservative variable ˜

Yefor-

ward in time. At this point, the updated conservative

variables would be known at all grid points except the

outer boundary. The next step is to recover the primi-

tive variables given these evolved conservative variables

(see Sec. III B). After the primitives have been recovered,

they are checked for physicality and marginally modiﬁed

if they are outside of their physical ranges [83]. For ex-

ample, in the case of the electron fraction, we check that

Ye,lower ≤Ye≤Ye,upper,(19)

where Ye,lower (Ye,upper) corresponds to the lowest (high-

est) value for Yeavailable in an EOS table. Next, outer

boundary conditions are placed on the recovered primi-

tives to ﬁll the necessary 3 ghost-zones in each direction.

We apply zero-derivative outﬂow outer boundary condi-

tions as described in [56]. Up to this point, the primitives

Pare known at the new time step on all grid points. The

ﬁnal step we take is to recompute the conservatives on all

grid points using Eq. (10) for consistency between Pand

C, and the evolution algorithm is allowed to proceed.

B. Conservative-to-primitive solvers

At every step of the evolution an inversion from the

evolved conservative variables Cto the physical primi-

tive variables Pis required to know the state of the ﬂuid.

Eq. (10) presents a system of non-linear algebraic expres-

sions which can be solved for 9 relevant ﬂuid variables

(ρb,vi,Bi,Ye, and either h, or ). These 9 variables,

along with an EOS, in turn provide all of the informa-

tion required to determine P, along with other ﬂuid vari-

ables of interest. For example, a solution to Eq. (10) can

provide the 5 main variables (ρ0,vi,) and, trivially, Bi

and Ye. We can then determine the remaining variable,

Pwith the use of an EOS and incidentally obtain infor-

mation on other physical variables such as the tempera-

ture Tand speciﬁc entropy sb. As we require a solution

for 5 main variables in order to determine P, the primi-

tives inversion problem is fundamentally a non-trivial 5D

problem that cannot be solved analytically. 5D schemes

which solve Eq. (10) were originally implemented in early

MHD codes such as HARM [87]. However, these schemes

were eventually found to be ineﬃcient and inaccurate,

which led to the development of methods which solve for

only two auxiliary variables [88] and thereby reduce the

dimensionality of the problem to 2D. The dimensional-

ity of the problem can be further reduced to 1D; modern

1D algorithms which provide reliable and eﬃcient solu-

tions have been developed [17,89] and are widely used

in GRMHD codes.

In order to consider strongly magnetized systems

which include realistic descriptions of the dense mat-

ter EOS, we implement state-of-the-art conservative-to-

primitive solvers within MIL. Our implementation in-

cludes porting the solvers discussed in [84] for use in

the EinsteinToolkit, packaged within a new thorn

ConservativeToPrimitive which can interface with MIL

and its associated thorns, but also works as a standalone

thorn which can be used with other GRMHD codes that

operate within the Cactus infrastructure. We focus on

a subset of the solvers implemented in [84], due to their

reliability, speed, and algorithmic similarity to the origi-

nal solvers used in OIL (see [90] for another possible ap-

proach). In particular we focus on the 2D solver of [88]

(which we label Noble), the 1D solver of [17] (which

we label Palenzuela), and the 1D solver of [89] (which

we label Newman). We note that the Noble solver uses

the same algorithm as the solvers in the original ver-

sion of IllinoisGRMHD and that other GRMHD codes

with realistic EOS capability rely on the Palenzuela

algorithm [58,91,92]. We refer the reader to [84] for

a review of the algorithms used in each solver within

ConservativeToPrimitive1.

We employ a set of preliminary tests to conﬁrm that

our port of these solvers to the EinsteinToolkit be-

haves as intended. In particular, we test the aforemen-

tioned solvers using the same tests as [84] where a set of

primitives Pare initialized, randomly perturbed, used to

calculate a set of conservatives C, and ﬁnally recovered

into a new set P’. The primitives recovery for each solver

is then assessed by considering the relative error between

the original set Pand the recovered set P0(along with

other diagnostics including the number of interpolation

calls to the EOS table and number of algorithm itera-

tions). In these tests, a subset of the primitives is varied

over the physically allowed range while holding others

constant. We focus on the case of the LS220 realistic,

ﬁnite temperature, tabulated EOS [85,86] and employ

tests where we prescribe the ratio of magnetic to ﬂuid

pressure Pmag/P =b2/(2P)=0.001, the Lorentz factor

W= 2, and the electron fraction Ye= 0.1, while scan-

ning over the allowed range of rest mass density ρband

1We point out that [84] appears to have typographical errors in

the algorithm description for the Newman solver, when compared

to the original paper of [89]. In particular, there is a diﬀerence

in the calculation of the auxiliary variable M2(see Eq. (47)

of [84] as compared to Eq. (4.7) of [89]), where [89] correctly

calculates it as M2=mimi=˜

Si˜

Si, where ˜

Siis the conservative

variable associated with the momentum density which appears

in the left-hand-side of Eq. (15). Eq. (47) of [84] inconsistently

calculates this variable as M2= (Bivi)2/√γ, where γis the

determinant of the 3-metric γij ,Biis the magnetic ﬁeld, and

viis the ﬂuid 3-velocity. We also note that the ﬁrst term of

Eq. (55) in [84] misses a factor of the auxiliary variable a, when

compared to the analogous Eq. (5.11) in [89]. We point out that

despite these typographical errors, the numerical implementation

of the Newman solver within the code of [84] is consistent with the

correct algorithmic steps presented in [89].

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QuantifyinguncertaintiesingeneralrelativisticmagnetohydrodynamiccodesPedroL.Espino,1,2GabrieleBozzola,3andVasileiosPaschalidis3,41DepartmentofPhysics,ThePennsylvaniaStateUniversity,UniversityPark,PA16802,USA2DepartmentofPhysics,UniversityofCalifornia,Berkeley,CA94720,USA3DepartmentofAstronomy,Univer...

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Quantifying uncertainties in general relativistic magnetohydrodynamic codes Pedro L. Espino1 2Gabriele Bozzola3and Vasileios Paschalidis3 4 1Department of Physics The Pennsylvania State University University Park PA 16802 USA.pdf

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Quantifying uncertainties in general relativistic magnetohydrodynamic codes Pedro L. Espino1 2Gabriele Bozzola3and Vasileios Paschalidis3 4 1Department of Physics The Pennsylvania State University University Park PA 16802 USA

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