Flux-conservative Form Research Articles

General-relativistic hydrodynamics of non-perfect fluids: 3+1 conservative formulation and application to viscous black hole accretion

ABSTRACT We consider the relativistic hydrodynamics of non-perfect fluids with the goal of determining a formulation that is suited for numerical integration in special-relativistic and general-relativistic scenarios. To this end, we review the various formulations of relativistic second-order dissipative hydrodynamics proposed so far and present in detail a particular formulation that is fully general, causal, and can be cast into a 3+1 flux-conservative form, as the one employed in modern numerical-relativity codes. As an example, we employ a variant of this formulation restricted to a relaxation-type equation for the bulk viscosity in the general-relativistic magnetohydrodynamics code bhac. After adopting the formulation for a series of standard and non-standard tests in 1+1-dimensional special-relativistic hydrodynamics, we consider a novel general-relativistic scenario, namely, the stationary, spherically symmetric, viscous accretion on to a black hole. The newly developed solution – which can exhibit even considerable deviations from the inviscid counterpart – can be used as a testbed for numerical codes simulating non-perfect fluids on curved backgrounds.

MOCMC: Method of Characteristics Moment Closure, a Numerical Method for Covariant Radiation Magnetohydrodynamics

Abstract We present a conservative numerical method for radiation magnetohydrodynamics with frequency-dependent full transport in stationary spacetimes. This method is stable and accurate for both large and small optical depths and radiation pressures. The radiation stress–energy tensor is evolved in flux-conservative form, and closed with a swarm of samples that each transport a multigroup representation of the invariant specific intensity along a null geodesic. In each zone, the enclosed samples are used to efficiently construct a Delaunay triangulation of the unit sphere in the comoving frame, which in turn is used to calculate the Eddington tensor, average source terms, and adaptively refine the sample swarm. Radiation four-forces are evaluated in the moment sector in a semi-implicit fashion. The radiative transfer equation is solved in invariant form deterministically for each sample. Since each sample carries a discrete representation of the full spectrum, the cost of evaluating the transport operator is independent of the number of frequency groups, representing a significant reduction of algorithmic complexity for transport in frequency-dependent problems. The major approximation we make in this work is performing scattering in an angle-averaged way. Local adaptivity in samples also makes this scheme more amenable to nonuniform meshes than a traditional Monte Carlo method. We describe the method and present results on a suite of test problems. We find that Method of Characteristics Moment Closure converges at least as ∼N −1, rather than the canonical Monte Carlo N −1/2, where N is the number of samples per zone.

General relativistic hydrodynamics in curvilinear coordinates

In this paper we report on what we believe is the first successful implementation of relativistic hydrodynamics, coupled to dynamical spacetimes, in spherical polar coordinates without symmetry assumptions. We employ a high-resolution shock-capturing scheme, which requires that the equations be cast in flux-conservative form. One example of such a form is the :Valencia" formulation, which has been adopted in numerous applications, in particular in Cartesian coordinates. Here we generalize this formulation to allow for a reference-metric approach, which provides a natural framework for calculations in curvilinear coordinates. In spherical polar coordinates, for example, it allows for an analytical treatment of the singular r and sin(\theta) terms that appear in the equations. We experiment with different versions of our generalized Valencia formulation in numerical implementations of relativistic hydrodynamics for both fixed and dynamical spacetimes. We consider a number of different tests -- non-rotating and rotating relativistic stars, as well as gravitational collapse to a black hole -- to demonstrate that our formulation provides a promising approach to performing fully relativistic astrophysics simulations in spherical polar coordinates.

BSSN equations in spherical coordinates without regularization: spherically symmetric spacetimes

Brown recently introduced a covariant formulation of the BSSN equations which is well suited for curvilinear coordinate systems. This is particularly desirable as many astrophysical phenomena are symmetric with respect to the rotation axis or are such that curvilinear coordinates adapt better to their geometry. We show results from a newly developed numerical code solving the BSSN equations in spherical symmetry and the general relativistic hydrodynamic equations written in flux-conservative form. A key feature of the code is that uses a second-order partially implicit Runge-Kutta method to integrate the evolution equations, and does not need a regularization algorithm at the origin. We discuss a number of tests to assess the accuracy, numerical stability and expected convergence of the code.

BSSN equations in spherical coordinates without regularization: Vacuum and nonvacuum spherically symmetric spacetimes

Brown [Phys. Rev. D 79, 104029 (2009)] has recently introduced a covariant formulation of the BSSN equations which is well suited for curvilinear coordinate systems. This is particularly desirable as many astrophysical phenomena are symmetric with respect to the rotation axis or are such that curvilinear coordinates adapt better to their geometry. However, the singularities associated with such coordinate systems are known to lead to numerical instabilities unless special care is taken (e.g., regularization at the origin). Cordero-Carri\'on will present a rigorous derivation of partially implicit Runge-Kutta methods in forthcoming papers, with the aim of treating numerically the stiff source terms in wavelike equations that may appear as a result of the choice of the coordinate system. We have developed a numerical code solving the BSSN equations in spherical symmetry and the general relativistic hydrodynamic equations written in flux-conservative form. A key feature of the code is that it uses a second-order partially implicit Runge-Kutta method to integrate the evolution equations. We perform and discuss a number of tests to assess the accuracy and expected convergence of the code---namely a pure gauge wave, the evolution of a single black hole, the evolution of a spherical relativistic star in equilibrium, and the gravitational collapse of a spherical relativistic star leading to the formation of a black hole. We obtain stable evolutions of regular spacetimes without the need for any regularization algorithm at the origin.

A numerical model of wave overtopping on seadikes

This paper describes the development of a numerical model for wave overtopping on seadikes. The model is based on the flux-conservative form of the nonlinear shallow water equations (NLSW) solved with a high order total variation diminishing (TVD), Roe-type scheme. The goal is to reliably predict the hydrodynamics of wave overtopping on the dike crest and along the inner slope, necessary for the breach modelling of seadikes. Besides the mean overtopping rate, the capability of simulating individual overtopping events is also required. It is shown theoretically that the effect of wave breaking through the drastic motion of surface rollers in the surfzone is not sufficiently described by the conventional nonlinear shallow water equations, neglecting wave setup from the mean water level and thus markedly reducing the model predictive capacity for wave overtopping. This is significantly improved by including an additional source term associated with the roller energy dissipation in the depth-averaged momentum equation. The developed model has been validated against four existing laboratory datasets of wave overtopping on dikes. The first two sets are to validate the roller term performance in improving the model prediction of wave overtopping of breaking waves. The last two sets are to test the model performance under more complex but realistic hydraulic and slope geometric conditions. The results confirm the merit of the supplemented roller term and also demonstrate that the model is robust and reliable for the prediction of wave overtopping on seadikes.

Nada: A new code for studying self-gravitating tori around black holes

We present a new two-dimensional numerical code called $\mathtt{N}\mathtt{a}\mathtt{d}\mathtt{a}$ designed to solve the full Einstein equations coupled to the general relativistic hydrodynamics equations. The code is mainly intended for studies of self-gravitating accretion disks (or tori) around black holes, although it is also suitable for regular spacetimes. Concerning technical aspects the Einstein equations are formulated and solved in the code using a formulation of the standard $3+1$ Arnowitt-Deser-Misner canonical formalism system, the so-called Baumgarte-Shapiro Shibata-Nakamura approach. A key feature of the code is that derivative terms in the spacetime evolution equations are computed using a fourth-order centered finite difference approximation in conjunction with the Cartoon method to impose the axisymmetry condition under Cartesian coordinates (the choice in $\mathtt{N}\mathtt{a}\mathtt{d}\mathtt{a}$), and the puncture/moving puncture approach to carry out black hole evolutions. Correspondingly, the general relativistic hydrodynamics equations are written in flux-conservative form and solved with high-resolution, shock-capturing schemes. We perform and discuss a number of tests to assess the accuracy and expected convergence of the code, namely, (single) black hole evolutions, shock tubes, and evolutions of both spherical and rotating relativistic stars in equilibrium, the gravitational collapse of a spherical relativistic star leading to the formation of a black hole. In addition, paving the way for specific applications of the code, we also present results from fully general relativistic numerical simulations of a system formed by a black hole surrounded by a self-gravitating torus in equilibrium.

Flux-conservative thermodynamic equations in a mass-weighted framework

ABSTRACTA set of equations, ready for discretization, is presented for the purely thermodynamic part of atmospheric energetics along a vertical column. Considerations of kinetic energy budgets and detailed turbulence laws are left for further study. The equations are derived in a total mass-based framework, both for the vertical coordinate system and for the conservation laws. This results in the use of the full barycentric velocity as the vector of advection. Under these conditions, the equations are derived from first principles on the basis of an a priori defined set of simplifying hypotheses. The originality of the resulting set of equations is twofold. First, even in the presence of a full prognostic treatment of cloud and precipitation processes, there exists a flux-conservative form for all relevant budgets, including that of the thermodynamic equation. Secondly, the form of the state law that is obtained for the multiphase system allows the flux-conservative form to be kept when going from the hydrostatic primitive equations system to the fully compressible system and projecting then the heat source/sink on both temperature and pressure tendencies.

Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes.

Gravitational waves from relativistic rotational core collapse in axisymmetry

We present results from simulations of axisymmetric relativistic rotational core collapse. The main objective of our investigation is to compute the waveforms of gravitational radiation emitted in such events, extending previous Newtonian simulations to relativity. The general relativistic hydrodynamic equations are formulated in flux-conservative form and solved using a high-resolution shock-capturing scheme. The Einstein equations are solved assuming a conformally flat 3-metric and the quadrupole formula is used to extract waveforms of the gravitational radiation emitted during the collapse. A comparison of our results with those of Newtonian simulations shows that gravitational wave amplitudes agree within 30%. Surprisingly, in some cases, relativistic effects actually diminish the amplitude of the gravitational wave signal. We further find that the parameter range of models suffering multiple coherent bounces due to centrifugal forces is considerably smaller than in Newtonian simulations.

Gravitational Waves from Relativistic Rotational Core Collapse

We present results from simulations of axisymmetric relativistic rotational core collapse. The general relativistic hydrodynamic equations are formulated in flux-conservative form and solved using a high-resolution shock-capturing scheme. The Einstein equations are solved assuming a conformally flat 3-metric, and the quadrupole formula is used to extract waveforms of the gravitational radiation emitted during the collapse. A comparison of our results with those of Newtonian simulations shows that the wave amplitudes agree within 30%. Surprisingly, in some cases, relativistic effects actually diminish the amplitude of the gravitational wave signal. We further find that the parameter range of models suffering multiple coherent bounces due to centrifugal forces is considerably smaller than in Newtonian simulations.

Causal differencing of flux-conservative equations applied to black hole spacetimes

We give a general scheme for finite-differencing partial differential equations in flux-conservative form to second order, with a stencil that can be arbitrarily tilted with respect to the numerical grid, parametrized by a `tilt' vector field . This can be used to centre the numerical stencil on the physical lightcone, by setting , where is the usual shift vector in the 3 + 1 split of spacetime, but other choices of the tilt may also be useful. We apply this `causal differencing' algorithm to the Bona-Massó equations, a hyperbolic and flux-conservative form of the Einstein equations, and demonstrate long-term stable causally correct evolutions of single black hole systems in spherical symmetry.

How to avoid artificial boundaries in the numerical calculation of black hole spacetimes

This is the first of a series of papers describing a numerical implementation of the conformally rescaled Einstein equation, an implementation designed to calculate asymptotically flat spacetimes, especially spacetimes containing black holes. We derive the new first-order time evolution equations to be used in the scheme. These time evolution equations can either be written in symmetric hyperbolic or in flux-conservative form. Since the conformally rescaled Einstein equation, also called the conformal field equations, formally allow us to place the grid boundaries outside the physical spacetime, we can modify the equations near the grid boundaries and obtain a consistent and stable discretization. Even if we calculate spacetimes containing black holes, there is no need to introduce artificial boundaries in the physical spacetime, which then would complicate, influence or even exclude the computation of certain spacetime regions.

Flux-corrected pseudospectral method for scalar hyperbolic conservation laws

The pseudospectral method has under-used advantages in problems involving shocks and discontinuities. These emerge from superior accuracy in phase and group velocities as compared to finite difference schemes of all orders. Dispersion curves for finite difference schemes suggest that group velocity error typically outranks Gibbs' error as a cause of numerical oscillation. A flux conservative form of the pseudospectral method is derived for compatibility with flux limiters used to preserve monotonicity. The resulting scheme gives high quality results in linear advection and shock formation/propagation examples.