Convergence and Divergence of the Solutions of a Neutral Difference Equation

We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form Δ[()+(())]+()(())=0, where () is a general retarded argument, () is a general deviated argument (retarded or advanced), ∈ℝ, (())≥0 is a sequence of positive real numbers such that ()≥, ∈ℝ+, and Δ denotes the forward difference operator Δ()=(+1)−(). Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to

An Explicit Scheme for Incorporating Ambipolar Diffusion in a Magnetohydrodynamics Code

We describe a method for incorporating ambipolar diffusion in the strong coupling approximation into a multidimensional magnetohydrodynamics code based on the total variation diminishing scheme. Contributions from ambipolar diffusion terms are included by explicit finite difference operators in a fully unsplit way, maintaining second order accuracy. The divergence-free condition of magnetic fields is exactly ensured at all times by a flux-interpolated constrained transport scheme. The super time stepping method is used to accelerate the timestep in high resolution calculations and/or in strong ambipolar diffusion. We perform two test problems, the steady-state oblique C-type shocks and the decay of Alfv\'en waves, confirming the accuracy and robustness of our numerical approach. Results from the simulations of the compressible MHD turbulence with ambipolar diffusion show the flexibility of our method as well as its ability to follow complex MHD flows in the presence of ambipolar diffusion. These simulations show that the dissipation rate of MHD turbulence is strongly affected by the strength of ambipolar diffusion.Comment: 25 pages, 5 figures, ApJS accepte

A Simflowny-based high-performance 3D code for the generalized induction equation

In the interior of neutron stars, the induction equation regulates the long-term evolution of the magnetic fields by means of resistivity, Hall dynamics and ambipolar diffusion. Despite the apparent simplicity and compactness of the equation, the dynamics it describes is not trivial and its understanding relies on accurate numerical simulations. While a few works in 2D have reached a mature stage and a consensus on the general dynamics at least for some simple initial data, only few attempts have been performed in 3D, due to the computational costs and the need for a proper numerical treatment of the intrinsic non-linearity of the equation. Here, we carefully analyze the general induction equation, studying its characteristic structure, and we present a new Cartesian 3D code, generated by the user-friendly, publicly available {\em Simflowny} platform. The code uses high-order numerical schemes for the time and spatial discretization, and relies on the highly-scalable {\em SAMRAI} architecture for the adaptive mesh refinement. We present the application of the code to several benchmark tests, showing the high order of convergence and accuracy achieved and the capabilities in terms of magnetic shock resolution and three-dimensionality. This paper paves the way for the applications to a realistic, 3D long-term evolution of neutron stars interior and, possibly, of other astrophysical sources.Comment: 23 pages, 13 figures. In pres

A three-dimensional numerical method for modelling weakly ionized plasmas

Astrophysical fluids under the influence of magnetic fields are often subjected to single-fluid or two-fluid approximations. In the case of weakly ionized plasmas however, this can be inappropriate due to distinct responses from the multiple constituent species to both collisional and non-collisional forces. As a result, in dense molecular clouds and proto-stellar accretion discs for instance, the conductivity of the plasma may be highly anisotropic leading to phenomena such as Hall and ambipolar diffusion strongly influencing the dynamics. Diffusive processes are known to restrict the stability of conventional numerical schemes which are not implicit in nature. Furthermore, recent work establishes that a large Hall term can impose an additional severe stability limit on standard explicit schemes. Following a previous paper which presented the one-dimensional case, we describe a fully three-dimensional method which relaxes the normal restrictions on explicit schemes for multifluid processes. This is achieved by applying the little known Super TimeStepping technique to the symmetric (ambipolar) component of the evolution operator for the magnetic field in the local plasma rest-frame, and the new Hall Diffusion Scheme to the skew-symmetric (Hall) component.Comment: 13 pages, 9 figures, accepted for publication in MNRA