Searcharxiv⌕ Search

arXiv subjects

Oscar A. Reula

Publications and source records attributed to Oscar A. Reula.

At least 19 recordsLinked to original sources
arXiv2023

Numerical simulations of divergence-type theories for conformal dissipative fluids

We present the first numerical simulations of the symmetric--hyperbolic theory for conformal dissipative relativistic fluids developed in [1]. In this theory, the information of the fluid dynamics is encoded in a scalar generating function which depends on three free parameters. By adapting the WENO-Z high-resolution shock-capturing central scheme, we show numerical solutions restricted to planar symmetry in Minkowski spacetime, from two qualitatively different initial data: a smooth bump and a discontinuous step. We perform a detailed exploration of the effect of the different parameters of the theory, and numerically assess the constitutive relations associated with the shear viscosity by analyzing the entropy production rate when shocks are produced.

gr-qc↗
arXiv2019

Generic instabilities in the relativistic Chapman-Enskog heat conduction law

We address the well-posedness of the Cauchy problem corresponding to the relativistic fluid equations, when coupled with the heat-flux constitutive relation arising within the relativistic Chapman-Enskog procedure. The resulting system of equations is shown to be non hyperbolic, by considering general perturbations over the whole set of equations written with respect to a generic time direction. The obtained eigenvalues are not purely imaginary and their real part grows without bound as the wave-number increases. Unlike Eckart's theory, this instability is not present when the time direction is aligned with the fluid's direction. However, since in general the fluid velocity is not surface-forming, the instability can only be avoided in the particular case where no rotation is present.

gr-qc↗
arXiv2017

A hyperbolic theory of relativistic conformal dissipative fluids

We develop a complete description of the class of conformal relativistic dissipative fluids of divergence form, following the formalism carried out by Geroch, Lindblom and Pennisi. This type of theories is fully described in terms of evolution variables whose dynamics is governed by total divergence-type conservation laws. Specifically, we give a characterization of the whole family of conformal fluids in terms of a single master scalar function defined up to second order corrections in dissipative effects, which we explicitly find in general form. This allows us to identify the equilibrium states of the theory, as well as to derive constitutive relations and a Fourier-like law for the corresponding first-order theory heat flux. Finally, we show that among this class of theories-- and near equilibrium configurations-- there exist symmetric hyperbolic ones, implying that for them one can define well posed initial value problems.

gr-qc↗
arXiv2017

On the ill posedness of Force-Free Electrodynamics in Euler Potentials

We prove that the initial value problem for Force-free Electrodynamics in Euler variables is not well posed. We establish this result by showing that a well-posedness criterion provided by Kreiss fails to hold for this theory, and using a theorem provided by Strang. To show the nature of the problem we display a particular bounded (in Sobolev norms) sequence of initial data for the Force-free equations such that at any given time as close to zero as one wishes, the corresponding evolution sequence is not bounded. Thus, the Force-free evolution is non continuous in that norm with respect to the initial data. We furthermore prove that this problem is also ill-posed in the Leray-Ohya sense.

gr-qc↗
arXiv2011

Numerical treatment of interfaces in Quantum Mechanics

In this article we develop a numerical scheme to deal with interfaces between touching numerical grids when solving Schrö{}dinger equation. In order to pass the information among grids we use the values of the fields only at the contact point between them. Surprisingly we obtain a convergent methods which is third order accurate with respect to the spatial resolution. In test cases, at the minimal resolution needed to describe correctly the waves, the error of this approximation is similar to that of a homogeneous (centered differences everywhere) scheme with three points stencil, that is a sixth order finite difference operator. The semi-discrete approximation preserves the norm and uses standard finite difference operators satisfying summation by parts. For the time integrator we use a semi-implicit IMEX Runge Kutta method.

quant-ph↗
arXiv2004

Strongly hyperbolic second order Einstein's evolution equations

BSSN-type evolution equations are discussed. The name refers to the Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution equations, without introducing the conformal-traceless decomposition but keeping the three connection functions and including a densitized lapse. It is proved that a pseudo-differential first order reduction of these equations is strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner evolution equations are found to be weakly hyperbolic. In both cases, the positive densitized lapse function and the spacelike shift vector are arbitrary given fields. This first order pseudodifferential reduction adds no extra equations to the system and so no extra constraints.

gr-qc↗
arXiv2001

On free evolution of self gravitating, spherically symmetric waves

We perform a numerical free evolution of a selfgravitating, spherically symmetric scalar field satisfying the wave equation. The evolution equations can be written in a very simple form and are symmetric hyperbolic in Eddington-Finkelstein coordinates. The simplicity of the system allow to display and deal with the typical gauge instability present in these coordinates. The numerical evolution is performed with a standard method of lines fourth order in space and time. The time algorithm is Runge-Kutta while the space discrete derivative is symmetric (non-dissipative). The constraints are preserved under evolution (within numerical errors) and we are able to reproduce several known results.

gr-qc↗
arXiv1999

Well-posed forms of the 3+1 conformally-decomposed Einstein equations

We show that well-posed, conformally-decomposed formulations of the 3+1 Einstein equations can be obtained by densitizing the lapse and by combining the constraints with the evolution equations. We compute the characteristics structure and verify the constraint propagation of these new well-posed formulations. In these formulations, the trace of the extrinsic curvature and the determinant of the 3-metric are singled out from the rest of the dynamical variables, but are evolved as part of the well-posed evolution system. The only free functions are the lapse density and the shift vector. We find that there is a 3-parameter freedom in formulating these equations in a well-posed manner, and that part of the parameter space found consists of formulations with causal characteristics, namely, characteristics that lie only within the lightcone. In particular there is a 1-parameter family of systems whose characteristics are either normal to the slicing or lie along the lightcone of the evolving metric.

gr-qc↗
arXiv1999

On the dynamics of Einstein's equations in the Ashtekar formulation

We study the dynamics of Einstein's equations in Ashtekar's variables from the point of view of the theory of hyperbolic systems of evolution equations. We extend previous results and show that by a suitable modification of the Hamiltonian vector flow outside the sub-manifold of real and constrained solutions, a symmetric hyperbolic system is obtained for any fixed choice of lapse-shift pair, without assuming the solution to be a priori real. We notice that the evolution system is block diagonal in the pair $(σ^a,A_b)$, and provide explicit and very simple formulae for the eigenvector-eigenvalue pairs in terms of an orthonormal tetrad with one of its components pointing along the propagation direction. We also analyze the constraint equations and find that when viewed as functions of the extended phase space they form a symmetric hyperbolic system on their own. We also provide simple formulae for its eigenvectors-eigenvalues pairs.

gr-qc↗
arXiv1999

A comment on gr-qc/9901053

In eprint gr-qc/9901053 Gen Yoneda and Hisa-aki Shinkai have made several claims disputing our results in gr-qc/9710004, and gr-qc/9804019. We show here that these claims are not correct.

gr-qc↗
arXiv1998

Einstein's Equations with Asymptotically Stable Constraint Propagation

We introduce a proposal to modify Einstein's equations by embedding them in a larger symmetric hyperbolic system. The additional dynamical variables of the modified system are essentially first integrals of the original constraints. The extended system of equations reproduces the usual dynamics on the constraint surface of general relativity, and therefore naturally includes the solutions to Einstein gravity. The main feature of this extended system is that, at least for a linearized version of it, the constraint surface is an attractor of the time evolution. This feature suggests that this system may be a useful alternative to Einstein's equations when obtaining numerical solutions to full, non-linear gravity.

gr-qc↗
arXiv1997

The Newtonian Limit on Asymptotically Null Foliations

By suitably re-scaling the conformal Einstein's equations we are able to apply recent results in the theory of PDE, and prove that they possess slow solutions in a future neighborhood of an initial surface reaching ${\cal I}^+$. The structure of the equations obtained allows to split (up to any given order) the initial data into those generating slow solutions, i.e., those driven by the sources, and those generating fast solutions, i.e., those which represent gravitational radiation with no relation to the sources. Thus effectively resulting in a proposal to prescribe initial data for solutions with no extra radiation up to the order needed for each given application.

gr-qc↗
arXiv1997

Fast and Slow solutions in General Relativity: The Initialization Procedure

We apply recent results in the theory of PDE, specifically in problems with two different time scales, on Einstein's equations near their Newtonian limit. The results imply a justification to Postnewtonian approximations when initialization procedures to different orders are made on the initial data. We determine up to what order initialization is needed in order to detect the contribution to the quadrupole moment due to the slow motion of a massive body as distinct from initial data contributions to fast solutions and prove that such initialization is compatible with the constraint equations. Using the results mentioned the first Postnewtonian equations and their solutions in terms of Green functions are presented in order to indicate how to proceed in calculations with this approach.

gr-qc↗
arXiv1997

Stability of quasi-linear hyperbolic dissipative systems

In this work we want to explore the relationship between certain eigenvalue condition for the symbols of first order partial differential operators describing evolution processes and the linear and nonlinear stability of their stationary solutions.

funct-an↗
arXiv1997

Global existence and exponential decay for hyperbolic dissipative relativistic fluid theories

We consider dissipative relativistic fluid theories on a fixed flat, compact, globally hyperbolic, Lorentzian manifold. We prove that for all initial data in a small enough neighborhood of the equilibrium states (in an appropriate Sobolev norm), the solutions evolve smoothly in time forever and decay exponentially to some, in general undetermined, equilibrium state. To prove this, three conditions are imposed on these theories. The first condition requires the system of equations to be symmetric hyperbolic, a fundamental requisite to have a well posed and physically consistent initial value formulation. The second condition is a generic consequence of the entropy law, and is imposed on the non principal part of the equations. The third condition is imposed on the principal part of the equations and it implies that the dissipation affects all the fields of the theory. With these requirements we prove that all the eigenvalues of the symbol associated to the system of equations of the fluid theory have strictly negative real parts, which in fact, is an alternative characterization for the theory to be totally dissipative. Once this result has been obtained, a straight forward application of a general stability theorem due to Kreiss, Ortiz, and Reula, implies the results above mentioned.

gr-qc↗
arXiv1996

First-order symmetric-hyperbolic Einstein equations with arbitrary fixed gauge

We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge-fixing.

gr-qc↗