Abstract. We have studied the stability in some anisotropic and isotropic matter configurations, by using the concept of cracking in General Relativity, which was conceived as an alternative and different approach to detect instabilities. It is based in the analysis of the radial forces that appear on the system after a perturbation capable of taking it out of its equilibrium state. We have studied the effects of local density perturbations on systems with "barotropic" equations of state. The considerations of the perturbations on the gradient of pressure lead to very different results than the previous works, and have found that not only anisotropic models may present cracking (or overturning), but it can also occur in isotropic matter configurations. IntroductionThe concepts of cracking and overturning were developed by Herrera and co-workers [1,2,3], with the aim of constituting an alternative way to determine instabilities on anisotropic matter configurations, and is complementary to the traditional formalism of Chandrasekhar [5,6,7]. The idea of studying instabilities around the appearance of cracking is very intuitive, it comes from the evaluation of the distribution of radial forces that appear on the system due to perturbations, and if there is a change of sign on this forces then it said that cracking occurs and we have an unstable configuration. The type of perturbation that are considered are capable of taking the system out of is equilibrium state so this distribution of forces may appear.In the first contributions where the stability was studied with the cracking criterion [1, 2, 3], were analysed independent and simultaneous perturbations on the density and of the anisotropy of some spherical solutions, were it was assumed that the perturbations did not change the gradient of pressure. In a more recent contribution [8], were studied perturbations of the density on systems governed by barotropic equations of state, were the anistotropy was affected though the equations of state and not independently, however the type of perturbations were constant since they did not change the gradient of pressure.In this paper, we will discuss how a different type of perturbations can be considered, following [8] we will also consider barotropic equations of state and will analyse perturbations of the density. The mean difference is that we want to consider local perturbations of the density, that are localized on a reduced region of the sphere. We will show the difference with the previous works when the effects of a local perturbation are considered. In section 2, we will describe the

We explore the influence of density fluctuations on isotropic and anisotropic configurations, extending the concept of cracking for general relativistic fluid spheres. This concept, conceived to describe the behaviour of anisotropic matter distributions just after its departure from equilibrium, could provide some insight on potential instabilities and future evolution of relativistic fluids. We have refined the idea of cracking, considering local fluctuations -represented by any function of compact support defined in a closed interval-and their effect on the state variables and their gradients through "barotropic" equations of state, P = P (ρ) and P ⊥ = P ⊥ (ρ). Under this approach it is found that both isotropic and anisotropic models could exhibit cracking (or overturning), and that previously crackable anisotropic models become uncrackable. *

Context. Thermal conductivity is one of the important mechanisms of heat transfer in the solar corona. Within the limit of strongly magnetized plasma, it is typically modeled by Spitzer's expression, where the heat flux is aligned with the magnetic field. Aims. This paper describes the implementation of heat conduction into the MANCHA3D code, with an aim of extending single-fluid magnetohydrodynamic (MHD) simulations from the upper convection zone into the solar corona. Methods. Two different schemes for modeling heat conduction are implemented in this work: (1) a standard scheme where a parabolic term is added to the energy equation and (2) a scheme where the hyperbolic heat flux equation is solved. Results. The first scheme limits the time step due to the explicit integration of a parabolic term, which makes the simulations computationally expensive. The second scheme solves the limitations on the time step by artificially limiting the heat conduction speed to computationally manageable values. The validation of both schemes was carried out with standard tests in one, two, and three spatial dimensions. Furthermore, we implemented the model for heat flux derived by Braginskii (1965) in its most general form, where the expression for the heat flux depends on the ratio of the collisional to cyclotron frequencies of the plasma and, therefore, on the magnetic field strength. Additionally, our implementation takes into account the heat conduction in parallel, perpendicular, and transverse directions, and provides the contributions from ions and electrons separately. The model recovers Spitzer's expression for parallel thermal conductivity within the strongly magnetized limit but also transitions smoothly between field-aligned conductivity and isotropic conductivity for regions with a low or null magnetic field. We describe the details of the implementation of Braginskii's thermal conductivity using a combination of the first scheme for the perpendicular and transverse directions and the second scheme for the parallel component. Furthermore, we estimated the thermal conductivities in a quiet-Sun model. In this model, we find that the perpendicular and transverse components for electrons and ions and the parallel component for ions might have some significance below the transition region. Above the transition region only the parallel component for ions might be important. Finally, we present a 2D test for heat conduction using realistic values of the solar atmosphere where we prove the robustness of the two schemes implemented and show that our adaptation of the hyperbolic treatment offers a great advantage over the computational cost of the simulations.

We present a new magnetohydrodynamic (MHD) code for the simulation of wave propagation in the solar atmosphere, under the effects of electrical resistivity -but not dominant-and heat transference in a uniform 3D grid. The code is based on the finite volume method combined with the HLLE and HLLC approximate Riemann solvers, which use different slope limiters like MINMOD, MC, and WENO5. In order to control the growth of the divergence of the magnetic field, due to numerical errors, we apply the Flux Constrained Transport method, which is described in detail to understand how the resistive terms are included in the algorithm. In our results, it is verified that this method preserves the divergence of the magnetic fields within the machine round-off error (∼ 1 × 10 −12 ). For the validation of the accuracy and efficiency of the schemes implemented in the code, we present some numerical tests in 1D and 2D for the ideal MHD. Later, we show one test for the resistivity in a magnetic reconnection process and one for the thermal conduction, where the temperature is advected by the magnetic field lines. Moreover, we display two numerical problems associated with the MHD wave propagation. The first one corresponds to a 3D evolution of a vertical velocity pulse at the photosphere-transition-corona region, while the second one consists in a 2D simulation of a transverse velocity pulse in a coronal loop.