35,910 research outputs found

    Trumpet Initial Data for Boosted Black Holes

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    We describe a procedure for constructing initial data for boosted black holes in the moving-punctures approach to numerical relativity that endows the initial time slice from the outset with trumpet geometry within the black hole interiors. We then demonstrate the procedure in numerical simulations using an evolution code from the Einstein Toolkit that employs 1+log slicing. The Lorentz boost of a single black hole can be precisely specified and multiple, widely separated black holes can be treated approximately by superposition of single hole data. There is room within the scheme for later improvement to re-solve (iterate) the constraint equations in the multiple black hole case. The approach is shown to yield an initial trumpet slice for one black hole that is close to, and rapidly settles to, a stationary trumpet geometry. Initial data in this new approach is shown to contain initial transient (or "junk") radiation that is suppressed by as much as two orders of magnitude relative to that in comparable Bowen-York initial data.Comment: 18 pages, 18 figure

    Hyperboloidal slices for the wave equation of Kerr-Schild metrics and numerical applications

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    We present new results from two open source codes, using finite differencing and pseudo-spectral methods for the wave equations in (3+1) dimensions. We use a hyperboloidal transformation which allows direct access to null infinity and simplifies the control over characteristic speeds on Kerr-Schild backgrounds. We show that this method is ideal for attaching hyperboloidal slices or for adapting the numerical resolution in certain spacetime regions. As an example application, we study late-time Kerr tails of sub-dominant modes and obtain new insight into the splitting of decay rates. The involved conformal wave equation is freed of formally singular terms whose numerical evaluation might be problematically close to future null infinity.Comment: 15 pages, 12 figure

    Three discontinuous Galerkin schemes for the anisotropic heat conduction equation on non-aligned grids

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    We present and discuss three discontinuous Galerkin (dG) discretizations for the anisotropic heat conduction equation on non-aligned cylindrical grids. Our most favourable scheme relies on a self-adjoint local dG (LDG) discretization of the elliptic operator. It conserves the energy exactly and converges with arbitrary order. The pollution by numerical perpendicular heat fluxes degrades with superconvergence rates. We compare this scheme with aligned schemes that are based on the flux-coordinate independent approach for the discretization of parallel derivatives. Here, the dG method provides the necessary interpolation. The first aligned discretization can be used in an explicit time-integrator. However, the scheme violates conservation of energy and shows up stagnating convergence rates for very high resolutions. We overcome this partly by using the adjoint of the parallel derivative operator to construct a second self-adjoint aligned scheme. This scheme preserves energy, but reveals unphysical oscillations in the numerical tests, which result in a decreased order of convergence. Both aligned schemes exhibit low numerical heat fluxes into the perpendicular direction. We build our argumentation on various numerical experiments on all three schemes for a general axisymmetric magnetic field, which is closed by a comparison to the aligned finite difference (FD) schemes of References [1,2

    Steady and Stable: Numerical Investigations of Nonlinear Partial Differential Equations

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    Excerpt: Mathematics is a language which can describe patterns in everyday life as well as abstract concepts existing only in our minds. Patterns exist in data, functions, and sets constructed around a common theme, but the most tangible patterns are visual. Visual demonstrations can help undergraduate students connect to abstract concepts in advanced mathematical courses. The study of partial differential equations, in particular, benefits from numerical analysis and simulation

    On the equilibrium morphology of systems drawn from spherical collapse experiments

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    We present a purely theoretical study of the morphological evolution of self-gravitating systems formed through the dissipationless collapse of N-point sources. We explore the effects of resolution in mass and length on the growth of triaxial structures formed by an instability triggered by an excess of radial orbits. We point out that as resolution increases, the equilibria shift, from mildly prolate, to oblate. A number of particles N ~= 100000 or larger is required for convergence of axial aspect ratios. An upper bound for the softening, e ~ 1/256, is also identified. We then study the properties of a set of equilibria formed from scale-free cold initial mass distributions, ro ~ r^-g with 0 <= g <= 2. Oblateness is enhanced for initially more peaked structures (larger values of g). We map the run of density in space and find no evidence for a power-law inner structure when g <= 3/2 down to a mass fraction <~0.1 per cent of the total. However, when 3/2 < g <= 2, the mass profile in equilibrium is well matched by a power law of index ~g out to a mass fraction ~ 10 per cent. We interpret this in terms of less-effective violent relaxation for more peaked profiles when more phase mixing takes place at the centre. We map out the velocity field of the equilibria and note that at small radii the velocity coarse-grained distribution function (DF) is Maxwellian to a very good approximation.Comment: 16 page
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