Scalable explicit implementation of anisotropic diffusion with Runge-Kutta-Legendre super-time-stepping

An important ingredient in numerical modelling of high temperature magnetised astrophysical plasmas is the anisotropic transport of heat along magnetic field lines from higher to lower temperatures.Magnetohydrodynamics (MHD) typically involves solving the hyperbolic set of conservation equations along with the induction equation. Incorporating anisotropic thermal conduction requires to also treat parabolic terms arising from the diffusion operator. An explicit treatment of parabolic terms will considerably reduce the simulation time step due to its dependence on the square of the grid resolution ($\Delta x$) for stability. Although an implicit scheme relaxes the constraint on stability, it is difficult to distribute efficiently on a parallel architecture. Treating parabolic terms with accelerated super-time stepping (STS) methods has been discussed in literature but these methods suffer from poor accuracy (first order in time) and also have difficult-to-choose tuneable stability parameters. In this work we highlight a second order (in time) Runge Kutta Legendre (RKL) scheme (first described by Meyer et. al. 2012) that is robust, fast and accurate in treating parabolic terms alongside the hyperbolic conversation laws. We demonstrate its superiority over the first order super time stepping schemes with standard tests and astrophysical applications. We also show that explicit conduction is particularly robust in handling saturated thermal conduction. Parallel scaling of explicit conduction using RKL scheme is demonstrated up to more than $10^4$ processors.Comment: 15 pages, 9 figures, incorporated comments from the referee. This version is now accepted for publication in MNRA

Modification of Angular Velocity by Inhomogeneous MRI Growth in Protoplanetary Disks

We have investigated evolution of magneto-rotational instability (MRI) in protoplanetary disks that have radially non-uniform magnetic field such that stable and unstable regions coexist initially, and found that a zone in which the disk gas rotates with a super-Keplerian velocity emerges as a result of the non-uniformly growing MRI turbulence. We have carried out two-dimensional resistive MHD simulations with a shearing box model. We found that if the spatially averaged magnetic Reynolds number, which is determined by widths of the stable and unstable regions in the initial conditions and values of the resistivity, is smaller than unity, the original Keplerian shear flow is transformed to the quasi-steady flow such that more flattened (rigid-rotation in extreme cases) velocity profile emerges locally and the outer part of the profile tends to be super-Keplerian. Angular momentum and mass transfer due to temporally generated MRI turbulence in the initially unstable region is responsible for the transformation. In the local super-Keplerian region, migrations due to aerodynamic gas drag and tidal interaction with disk gas are reversed. The simulation setting corresponds to the regions near the outer and inner edges of a global MRI dead zone in a disk. Therefore, the outer edge of dead zone, as well as the inner edge, would be a favorable site to accumulate dust particles to form planetesimals and retain planetary embryos against type I migration.Comment: 28 pages, 11figures, 1 table, accepted by Ap

Sparse non-negative super-resolution -- simplified and stabilised

The convolution of a discrete measure, $x=\sum_{i=1}^ka_i\delta_{t_i}$, with a local window function, $\phi(s-t)$, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources $\{a_i,t_i\}_{i=1}^k$ with an accuracy beyond the essential support of $\phi(s-t)$, typically from $m$ samples $y(s_j)=\sum_{i=1}^k a_i\phi(s_j-t_i)+\eta_j$, where $\eta_j$ indicates an inexactness in the sample value. We consider the setting of $x$ being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that $x$ is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. $\eta_j=0$, $m\ge 2k+1$ samples are available, and $\phi(s-t)$ generates a Chebyshev system. This is independent of how close the sample locations are and {\em does not rely on any regulariser beyond non-negativity}; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions $\hat{x}$ consistent with the samples within the bound $\sum_{j=1}^m\eta_j^2\le \delta^2$. Any such non-negative measure is within ${\mathcal O}(\delta^{1/7})$ of the discrete measure $x$ generating the samples in the generalised Wasserstein distance, converging to one another as $\delta$ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of $\phi(s-t)$ being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.Comment: 59 pages, 7 figure

Renormalization Group Flows from Gravity in Anti-de Sitter Space versus Black Hole No-Hair Theorems

Black hole no-hair theorems are proven using inequalities that govern the radial dependence of spherically symmetric configurations of matter fields. In this paper, we analyze the analogous inequalities for geometries dual to renormalization group flows via the AdS/CFT correspondence. These inequalities give much useful information about the qualitative properties of such flows. For Poincare invariant flows, we show that generic flows of relevant or irrelevant operators lead to singular geometries. For the case of irrelevant operators, this leads to an apparent conflict with Polchinski's decoupling theorem, and we offer two possible resolutions to this problem.Comment: 13 pages, 3 figures, harvmac, epsf, references and comments adde