4 the other thermodynamical quantities of the fluid, P = P ( ρ$_{o}$, ϵ ). High resolution shock capturing schemes (HRSC) are robust numerical methods for compressible fluid dy- namics. These methods, based on Godunov’s seminal work [37], are fundamentally based on expressing the fluid equations as integral conservation laws. To this end, we introduce conservative variables q = ( D, S$_{i}$, τ, B $^{i}$) $^{T}$, where where h = det( h$_{ij}$ ). The fluid equations can now be written in balance law form where f k are flux functions, and s are source terms. The fluid equations in this form are If the enthalpy can be expressed as a simple function of the pressure, as can be done for the ideal fluid and a generalized EOS with a cold nuclear component, then we can express the pressure as a function x . For the ideal fluid EOS used here, the enthalpy equation Using the definition of τ , we define a function that is identically zero The set of fluid equations, Eqs. (26-29), are used to evolve the conservative variables. However, these equa- tions also contain the primitive variables which neces- sitates a step in the evolution scheme which solves for the primitive variables in terms of the conservative ones. Given the primitive variables, the conservative variables are easily calculated from the algebraic expressions in Eqs. (24). Calculating the primitive variables from the conservative variables, however, is more delicate, as it re- quires the solution of a transcendental equation. To this end we first define the quantity x ≡ h$_{e}$W $^{2}$. We then write S 2 = S $^{i}$S$_{i}$ in terms of x and solve for W $^{2}$, obtaining where κ is a dimensional constant. We use the polytropic EOS only for setting initial data. where Γ is the constant adiabatic exponent. Nuclear mat- ter in neutron stars is relatively stiff, and we set Γ = 2 in this work. When the fluid flow is adiabatic, this EOS reduces to the well known polytropic EOS Finally, we close the system of fluid equations with an equation of state (EOS). We choose the ideal fluid EOS where c$_{r}$ = κ and we have allowed for different speeds than light by introducing the parameter c$_{h}$ . Except in the tests, we will use the original prescription (17) with c$_{h}$ = 1. Here $^{3}$Γ i jk is the Christoffel symbol associated with the 3-metric h$_{ij}$ , and K is the trace of the extrin- sic curvature, K = K $^{i}$$_{i}$. Notice that the aforementioned system is an extended version of the one employed in our earlier works [5, 27, 28] Here we have added additional terms toggled by the parameter ζ which allow for consid- ering an extension of the “eight-wave” formulation which, in the absence of the cleaning field Ψ, ensures the strong hyperbolicity of the system [36] –at the cost of introduc- ing derivative terms in the sources–. Furthermore, by setting ζ = 1, the propagation speeds of two constraint violating modes become non-vanishing and hence viola- tions are dragged along by the fluid’s velocity. This is numerically convenient as possible violations will propa- gate off the grid. D = W ρ$_{o}$, (21) and B i is both a primitive and conservative variable. In an asymptotically flat spacetime these quantities are conserved, and are related to the total energy, momen- tum, and, in the non-relativistic limit, the kinetic en- ergy, respectively. The quantities w = ( ρ$_{o}$, v $^{i}$, P, B$^{i}$) T are called the primitive variables in contrast to the con- servative variables. The fluid state can be specified using either set of variables, and both sets are required to write the MHD evolution equations. Anticipating the form of these equations, we also introduce the densitized con- served variables ˜ D = √$_{h D,}$ ˜ S$_{i}$ = √$_{h S}$$_{i}$$_{,}$ ˜ τ = √$_{h τ,}$ ˜ B i = √$_{h B}$ $^{i}$, (24) ∂$_{t}$ ˜ q + ∂$_{k}$ f $^{k}$(˜ q ) = s (˜ q ) , (25) W 2 = x $^{2}$( x + B $^{2}$)2 x $^{2}$( x + B $^{2}$)2 − x $^{2}$S2 − (2 x + B $^{2}$)( S$_{i}$B $^{i}$) $_{2}$. (33) f ( x ) = x − P − 1 2 [ ( S$_{i}$B $^{i}$) 2 x 2 + B 2 W 2 ] + B $^{2}$− τ − D = 0 . (34) h$_{e}$ = x W 2 = ρ$_{0}$ + ρ$_{0}$ϵ + P (35) P = κρ$_{o}$ $^{Γ}$, (32) P = (Γ − 1) ρ$_{o}$ϵ, (31) ∂$_{t}$ ˜ D + ∂$_{i}$ [ α ˜ D ( v i − β i α )] = 0 , (26) ∂$_{t}$ ˜ S$_{j}$ + ∂$_{i}$ [ α ( ˜ S$_{j}$ ( v i − β i α ) $_{+}$√$_{h P h}$ i j )] = α $^{3}$Γ i jk ( ˜ S$_{i}$v k $_{+}$√$_{h P h}$$_{i}$ k ) + ˜ S$_{a}$∂$_{j}$β a − ∂$_{j}$α (˜ τ + ˜ D ) ζα ( ˜ B$_{i}$W − 2 + v$_{i}$v$_{j}$ ˜ B $^{j}$) ∂$_{k}$ ˜ B $^{k}$, (27) S$_{i}$ = ( h$_{e}$W 2 + B $^{2}$) v$_{i}$ − ( B $^{j}$v$_{j}$ ) B$_{i}$, (22) τ = h$_{e}$W 2 + B 2 − P − 1 2 [ ( B $^{i}$v$_{i}$ ) 2 + B 2 W 2 ]$_{−}$ D. (23) ∂$_{t}$ ˜ τ + ∂$_{i}$ [ α ( ˜ S i − β i α ˜ τ − v i ˜ D )] = α [ K$_{ij}$ ˜ S $^{i}$vj $_{+}$√$_{h KP}$ − 1 α ˜ S $^{a}$∂$_{a}$ α ] , − ζαv$_{j}$ ˜ B $^{j}$∂$_{k}$ ˜ B k (28) ∂$_{t}$ ˜ B b + ∂$_{i}$ [ ˜ B b ( v i − β i α )$_{−}$ ˜ B i ( v b − β b α )] = − $_{α}$√$_{hh}$ $^{bi}$∂$_{i}$ Ψ − ζαv $^{i}$∂$_{j}$ ˜ B j (29) ∂$_{t}$ Ψ = − c$_{r}$α Ψ − c$_{h}$ α √$_{h}$ ∂$_{i}$ ˜ B i + ( β i − αv $^{i}$) ∂$_{i}$ Ψ . (30)