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      An Operator-Based Local Discontinuous Galerkin Method Compatible With the BSSN Formulation of the Einstein Equations

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          Abstract

          Discontinuous Galerkin Finite Element (DGFE) methods offer a mathematically beautiful, computationally efficient, and efficiently parallelizable way to solve hyperbolic PDEs. These properties make them highly desirable for numerical calculations in relativistic astrophysics and many other fields. The BSSN formulation of the Einstein equations has repeatedly demonstrated its robustness. The formulation is not only stable but allows for puncture-type evolutions of black hole systems. To-date no one has been able to solve the full (3+1)-dimensional BSSN equations using DGFE methods. This is partly because DGFE discretization often occurs at the level of the equations, not the derivative operator, and partly because DGFE methods are traditionally formulated for manifestly flux-conservative systems. By discretizing the derivative operator, we generalize a particular flavor of DGFE methods, Local DG methods, to solve arbitrary second-order hyperbolic equations. Because we discretize at the level of the derivative operator, our method can be interpreted as either a DGFE method or as a finite differences stencil with non-constant coefficients.

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          Most cited references58

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          The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

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            Total variation diminishing Runge-Kutta schemes

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              An Interior Penalty Finite Element Method with Discontinuous Elements

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                Author and article information

                Journal
                2016-03-31
                2016-04-14
                Article
                1604.00075
                735f8c74-c7f2-4ffb-ba7f-b4dc7bfaf2fd

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                83-08
                Clarified argument in section 2 and added a figure. Added a discussion on stability via summation-by-parts in the appendix and a discussion of consistency in section 2. Changed discussion of advection equation to discussion of wave equation. Result unchanged. Minor formatting and citation corrections. Fixed some typos. Added small clarifications
                gr-qc

                General relativity & Quantum cosmology
                General relativity & Quantum cosmology

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