Oscar Reula

Beyond ideal MHD: towards a more realistic modelling of relativistic astrophysical plasmas

Many astrophysical processes involving magnetic fields and quasi-stationary processes are well described when assuming the fluid as a perfect conductor. For these systems, the idealmagnetohydrodynamics (MHD) description captures the dynamics effectively and a number of well-tested techniques exist for its numerical solution. Yet, there are several astrophysical processes involving magnetic fields which are highly dynamical and for which resistive effects can play an important role. The numerical modelling of such non-ideal MHD flows is significantly more challenging as the resistivity is expected to change of several orders of magnitude across the flow and the equations are then either of hyperbolic‐parabolic nature or hyperbolic with stiff terms. We here present a novel approach for the solution of these relativistic resistive MHD equations exploiting the properties of implicit‐explicit (IMEX) Runge‐Kutta methods. By examining a number of tests, we illustrate the accuracy of our approach under a variety of conditions and highlight its robustness when compared with alternative methods, such as the Strang splitting. Most importantly, we show that our approach allows one to treat, within a unified framework, those regions of the flow which are both fluid-pressure dominated (such as in the interior of compact objects) and instead magnetic-pressure dominated (such as in their magnetospheres). In view of this, the approach presented here could find a number of applications and serve as a first step towards a more realistic modelling of relativistic astrophysical plasmas.

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.

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 pseudodifferential 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.

Multi-block simulations in general relativity: high-order discretizations, numerical stability and applications

The need to smoothly cover a computational domain of interest generically requires the adoption of several grids. To solve a given problem under this grid structure, one must ensure the suitable transfer of information among the different grids involved. In this work, we discuss a technique that allows one to construct finite-difference schemes of arbitrary high order which are guaranteed to satisfy linear numerical and strict stability. The method relies on the use of difference operators satisfying summation by parts and penalty terms to transfer information between the grids. This allows the derivation of semi-discrete energy estimates for problems admitting such estimates at the continuum. We analyse several aspects of this technique when used in conjunction with high-order schemes and illustrate its use in one-, two- and three-dimensional numerical relativity model problems with non-trivial topologies, including truly spherical black hole excision.

Well Posed Constraint-Preserving Boundary Conditions for the Linearized Einstein Equations

In this paper we address the problem of specifying boundary conditions for Einsteins equations when linearized around Minkowski space using the generalized Einstein-Christoffel symmetric hyperbolic system of evolution equations. The boundary conditions we work out guarantee that the constraints are satisfied provided they are satisfied on the initial slice and ensures a well posed initial-boundary value formulation. We consider the case of a manifold with a non-smooth boundary, as is the usual case of the cubic boxes commonly used in numerical relativity. The techniques discussed should be applicable to more general cases, as linearizations around more complicated backgrounds, and may be used to establish well posedness in the full non-linear case.

On the Newtonian limit of general relativity

We establish rigorous results about the Newtonian limit of general relativity by applying to it the theory of different time scales for non-linear partial differential equations as developed in [4, 1, 8]. Roughly speaking, we obtain a priori estimates for solutions to the Einsteins equations, an intermediate, but fundamental, step to show that given a Newtonian solution there exist continuous one-parameter families of solutions to the full Einsteins equations — the parameter being the inverse of the speed of light — which for a finite amount of time are close to the Newtonian solution. These one-parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions. This procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the Newtonian solution in any a priori given Sobolev norm. In some intuitive sense these relativistic solutions, by being close to the Newtonian one, have little extra radiation content (although, actually, this should be so only in the case of the characteristic initial data formulation along future directed light cones).Our results are local, in the sense that they do not include the treatment of asymptotic regions; global results are admittedly very important — in particular they would say how differentiable the solutions are with respect to the parameter — but their treatment would involve the handling of tools even more technical than the ones used here. On the other hand, this local theory is all that is needed for most problems of practical numerical computation.

Turbulent flows for relativistic conformal fluids in 2+1 dimensions

We demonstrate that relativistic conformal hydrodynamics in 2+1 dimensions displays a turbulent behaviour which cascades energy to longer wavelengths on both flat and spherical manifolds. Our motivation for this study is to understand the implications for gravitational solutions through the AdS/CFT correspondence. The observed behaviour implies gravitational perturbations of the corresponding black brane/black hole spacetimes (for sufficiently large scales/temperatures) will display a similar cascade towards longer wavelengths.

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.

The behavior of hyperbolic heat equations’ solutions near their parabolic limits

Standard energy methods are used to study the relation between the solutions of one parameter families of hyperbolic systems of equations describing heat propagation near their parabolic limits, which for these cases are the usual diffusive heat equation. In the linear case it is proven that given any solution to the hyperbolic equations there is always a solution to the diffusion equation which after a short time stays very close to it for all times. The separation between these solutions depends on the square of the ratio between the assumed very short decay time appearing in Cattaneo’s relation and the usual characteristic smoothing time (initial data dependent) of the limiting diffusive equation. The techniques used in the linear case can be readily used for nonlinear equations. As an example we consider the theories of heat propagation introduced by Coleman, Fabrizio, and Owen, and prove that near a solution to the limiting diffusive equation there is always a solution to the nonlinear hyperbolic equations for a time which usually is much longer than the decay time of the corresponding Cattaneo relation. An alternative derivation of the heat theories of divergence type, which are consistent with thermodynamic principles, is given as an appendix.

AMR, stability and higher accuracy

Efforts to achieve better accuracy in numerical relativity have so far focused either on implementing second-order accurate adaptive mesh refinement or on defining higher order accurate differences and update schemes. Here, we argue for the combination, that is a higher order accurate adaptive scheme. This combines the power that adaptive gridding techniques provide to resolve fine scales (in addition to a more efficient use of resources) together with the higher accuracy furnished by higher order schemes when the solution is adequately resolved. To define a convenient higher order adaptive mesh refinement scheme, we discuss a few different modifications of the standard, second-order accurate approach of Berger and Oliger. Applying each of these methods to a simple model problem, we find these options have unstable modes. However, a novel approach to dealing with the grid boundaries introduced by the adaptivity appears stable and quite promising for the use of high order operators within an adaptive framework.