An efficient quadrature rule on the cubed sphere

A new quadrature rule for functions defined on the sphere is introduced. The nodes are defined as the points of the Cubed Sphere. The associated weights are defined in analogy to the trapezoidal rule on each panel of the Cubed Sphere. The formula enjoys a symmetry property ensuring that a proportion of 7/8 of all Spherical Harmonics is integrated exactly. Based on the remaining Spherical Harmonics, it is possible to define modified weights giving an enhanced quad-rature rule. Numerical results show that the new quadrature is competitive with classical rules of the litterature. This second quadrature rule is believed to be of interest for applied mathematicians, physicists and engineers dealing with data located at the points of the Cubed Sphere

Application of the Cubed-Sphere Grid to Tilted Black-Hole Accretion Disks

In recent work we presented the first results of global general relativistic magnetohydrodynamic (GRMHD) simulations of tilted (or misaligned) accretion disks around rotating black holes. The simulated tilted disks showed dramatic differences from comparable untilted disks, such as asymmetrical accretion onto the hole through opposing "plunging streams" and global precession of the disk powered by a torque provided by the black hole. However, those simulations used a traditional spherical-polar grid that was purposefully underresolved along the pole, which prevented us from assessing the behavior of any jets that may have been associated with the tilted disks. To address this shortcoming we have added a block-structured "cubed-sphere" grid option to the Cosmos++ GRMHD code, which will allow us to simultaneously resolve the disk and polar regions. Here we present our implementation of this grid and the results of a small suite of validation tests intended to demonstrate that the new grid performs as expected. The most important test in this work is a comparison of identical tilted disks, one evolved using our spherical-polar grid and the other with the cubed-sphere grid. We also demonstrate an interesting dependence of the early-time evolution of our disks on their orientation with respect to the grid alignment. This dependence arises from the differing treatment of current sheets within the disks, especially whether they are aligned with symmetry planes of the grid or not.Comment: 15 pages, 11 figures, submitted to Ap

Strategies for the characteristic extraction of gravitational waveforms

We develop, test, and compare new numerical and geometrical methods for improving the accuracy of extracting waveforms using characteristic evolution. The new numerical method involves use of circular boundaries to the stereographic grid patches which cover the spherical cross sections of the outgoing null cones. We show how an angular version of numerical dissipation can be introduced into the characteristic code to damp the high frequency error arising form the irregular way the circular patch boundary cuts through the grid. The new geometric method involves use of the Weyl tensor component Psi4 to extract the waveform as opposed to the original approach via the Bondi news function. We develop the necessary analytic and computational formula to compute the O(1/r) radiative part of Psi4 in terms of a conformally compactified treatment of null infinity. These methods are compared and calibrated in test problems based upon linearized waves

Multi-patch methods in general relativistic astrophysics - I. Hydrodynamical flows on fixed backgrounds

Many systems of interest in general relativistic astrophysics, including neutron stars, accreting compact objects in X-ray binaries and active galactic nuclei, core collapse, and collapsars, are assumed to be approximately spherically symmetric or axisymmetric. In Newtonian or fixed-background relativistic approximations it is common practice to use spherical polar coordinates for computational grids; however, these coordinates have singularities and are difficult to use in fully relativistic models. We present, in this series of papers, a numerical technique which is able to use effectively spherical grids by employing multiple patches. We provide detailed instructions on how to implement such a scheme, and present a number of code tests for the fixed background case, including an accretion torus around a black hole.Comment: 26 pages, 20 figures. A high-resolution version is available at http://www.cct.lsu.edu/~bzink/papers/multipatch_1.pd

A Mixed Mimetic Spectral Element Model of the Rotating Shallow Water Equations on the Cubed Sphere

In a previous article [J. Comp. Phys. $\mathbf{357}$ (2018) 282-304], the mixed mimetic spectral element method was used to solve the rotating shallow water equations in an idealized geometry. Here the method is extended to a smoothly varying, non-affine, cubed sphere geometry. The differential operators are encoded topologically via incidence matrices due to the use of spectral element edge functions to construct tensor product solution spaces in $H(\mathrm{rot})$, $H(\mathrm{div})$ and $L_2$. These incidence matrices commute with respect to the metric terms in order to ensure that the mimetic properties are preserved independent of the geometry. This ensures conservation of mass, vorticity and energy for the rotating shallow water equations using inexact quadrature on the cubed sphere. The spectral convergence of errors are similarly preserved on the cubed sphere, with the generalized Piola transformation used to construct the metric terms for the physical field quantities

A Comparison of Two Shallow Water Models with Non-Conforming Adaptive Grids: classical tests

In an effort to study the applicability of adaptive mesh refinement (AMR) techniques to atmospheric models an interpolation-based spectral element shallow water model on a cubed-sphere grid is compared to a block-structured finite volume method in latitude-longitude geometry. Both models utilize a non-conforming adaptation approach which doubles the resolution at fine-coarse mesh interfaces. The underlying AMR libraries are quad-tree based and ensure that neighboring regions can only differ by one refinement level. The models are compared via selected test cases from a standard test suite for the shallow water equations. They include the advection of a cosine bell, a steady-state geostrophic flow, a flow over an idealized mountain and a Rossby-Haurwitz wave. Both static and dynamics adaptations are evaluated which reveal the strengths and weaknesses of the AMR techniques. Overall, the AMR simulations show that both models successfully place static and dynamic adaptations in local regions without requiring a fine grid in the global domain. The adaptive grids reliably track features of interests without visible distortions or noise at mesh interfaces. Simple threshold adaptation criteria for the geopotential height and the relative vorticity are assessed.Comment: 25 pages, 11 figures, preprin

Effective Inner Radius of Tilted Black Hole Accretion Disks

One of the primary means of determining the spin of an astrophysical black hole is by actually measuring the inner radius of a surrounding accretion disk and using that to infer the spin. By comparing a number of different estimates of the inner radius from simulations of tilted accretion disks with differing black-hole spins, we show that such a procedure can give quite wrong answers. Over the range 0 <= a/M <= 0.9, we find that, for moderately thick disks (H/r ~ 0.2) with modest tilt (15 degrees), the inner radius is nearly independent of spin. This result is likely dependent on tilt, such that for larger tilts, it may even be that the inner radius would increase with increasing spin. In the opposite limit, we confirm through numerical simulations of untilted disks that, in the limit of zero tilt, the inner radius recovers approximately the expected dependence on spin.Comment: 5 pages, 4 figures, accepted to ApJ Letter

An iterative algorithm for sparse and constrained recovery with applications to divergence-free current reconstructions in magneto-encephalography

We propose an iterative algorithm for the minimization of a $\ell_1$-norm penalized least squares functional, under additional linear constraints. The algorithm is fully explicit: it uses only matrix multiplications with the three matrices present in the problem (in the linear constraint, in the data misfit part and in penalty term of the functional). None of the three matrices must be invertible. Convergence is proven in a finite-dimensional setting. We apply the algorithm to a synthetic problem in magneto-encephalography where it is used for the reconstruction of divergence-free current densities subject to a sparsity promoting penalty on the wavelet coefficients of the current densities. We discuss the effects of imposing zero divergence and of imposing joint sparsity (of the vector components of the current density) on the current density reconstruction.Comment: 21 pages, 3 figure