786 research outputs found
ON THE GOTTLIEB POLYNOMIALS IN SEVERAL VARIABLES
In this study, we give some new properties of the Gottlieb polynomials in several variables. The results obtained here include various families of multilinear and multilateral generating functions, integral representation and recurrence relations for these polynomials. In addition, we derive a theorem giving certain families of bilateral generating functions for the multivariable Gottlieb polynomials and the generalized Lauricella functions. Finally, we get several results of this theorem
Spectral methods for the wave equation in second-order form
Current spectral simulations of Einstein's equations require writing the
equations in first-order form, potentially introducing instabilities and
inefficiencies. We present a new penalty method for pseudo-spectral evolutions
of second order in space wave equations. The penalties are constructed as
functions of Legendre polynomials and are added to the equations of motion
everywhere, not only on the boundaries. Using energy methods, we prove
semi-discrete stability of the new method for the scalar wave equation in flat
space and show how it can be applied to the scalar wave on a curved background.
Numerical results demonstrating stability and convergence for multi-domain
second-order scalar wave evolutions are also presented. This work provides a
foundation for treating Einstein's equations directly in second-order form by
spectral methods.Comment: 16 pages, 5 figure
Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators
We present a high-order accurate discontinuous Galerkin method for evolving
the spherically-reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system
expressed in terms of second-order spatial operators. Our multi-domain method
achieves global spectral accuracy and long-time stability on short
computational domains. We discuss in detail both our scheme for the BSSN system
and its implementation. After a theoretical and computational verification of
the proposed scheme, we conclude with a brief discussion of issues likely to
arise when one considers the full BSSN system.Comment: 35 pages, 6 figures, 1 table, uses revtex4. Revised in response to
referee's repor
Solving 3D relativistic hydrodynamical problems with WENO discontinuous Galerkin methods
Discontinuous Galerkin (DG) methods coupled to WENO algorithms allow high
order convergence for smooth problems and for the simulation of discontinuities
and shocks. In this work, we investigate WENO-DG algorithms in the context of
numerical general relativity, in particular for general relativistic
hydrodynamics. We implement the standard WENO method at different orders, a
compact (simple) WENO scheme, as well as an alternative subcell evolution
algorithm. To evaluate the performance of the different numerical schemes, we
study non-relativistic, special relativistic, and general relativistic
testbeds. We present the first three-dimensional simulations of general
relativistic hydrodynamics, albeit for a fixed spacetime background, within the
framework of WENO-DG methods. The most important testbed is a single TOV-star
in three dimensions, showing that long term stable simulations of single
isolated neutron stars can be obtained with WENO-DG methods.Comment: 21 pages, 10 figure
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