Related papers: A polytopal discrete de Rham scheme for the exteri…
Our work is about energy conserving fourth-order time discretizations of a three-field formulation of Maxwell's equations in conjunction with a spatial discretization using higher-order and compatible de Rham finite element spaces. Toward…
The purpose of this article is twofold. First we give a very robust method for proving sharp time decay estimates for the most classical three models of dispersive Partial Differential Equations, the wave, Klein-Gordon and Schr{\"o}dinger…
In this work, following the discrete de Rham (DDR) approach, we develop a discrete counterpart of a two-dimensional de Rham complex with enhanced regularity. The proposed construction supports general polygonal meshes and arbitrary…
We use an orthonormal frame approach to provide a general framework for the first order hyperbolic reduction of the Einstein equations coupled to a fairly generic class of matter models. Our analysis covers the special cases of dust and…
We provide in this paper the discrete equations of motion for the newtonian $n$-body problem deduced from the quantum calculus of variations (Q.C.V.) developed in \cite{Cre,CFT,RS1,RS2}. These equations are brought into the usual lagrangian…
Causal differencing has shown to be one of the promising and successful approaches towards excising curvature singularities from numerical simulations of black hole spacetimes. So far it has only been actively implemented in the ADM and…
We study the Einstein-Maxwell system of equations in spherically symmetric gravitational fields for static interior spacetimes. The condition for pressure isotropy is reduced to a recurrence equation with variable, rational coefficients. We…
Einstein's system of equations in the ADM decomposition involves two subsystems of equations: evolution equations and constraint equations. For numerical relativity, one typically solves the constraint equations only on the initial time…
We present the $d+1$ formulation of Einstein-scalar-Gauss-Bonnet (ESGB) theories in dimension $D=d+1$ and for arbitrary (spacelike or timelike) slicings. We first build an action which generalizes those of Gibbons-Hawking-York and Myers to…
We present a new derivation of the conservative form of the general relativistic Boltzmann equation and specialize it to the 3+1 metric. The resulting transport equation is intended for use in simulations involving numerical relativity,…
We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schr\"{o}dinger equation in…
This work proposes a novel variational approximation of partial differential equations on moving geometries determined by explicit boundary representations. The benefits of the proposed formulation are the ability to handle large…
A recent proposal equates the circuit complexity of a quantum gravity state with the gravitational action of a certain patch of spacetime. Since Einstein's equations follow from varying the action, it should be possible to derive them by…
We prove in this note that local geometric uniqueness holds true without loss of regularity for Einstein equations coupled with a large class of matter models. We thus extend the Planchon-Rodnianski uniqueness theorem for vacuum spacetimes.…
We present two semidiscretizations of the Camassa-Holm equation in periodic domains based on variational formulations and energy conservation. The first is a periodic version of an existing conservative multipeakon method on the real line,…
This work completes the task of solving locally the Einstein-Ashtekar equations for degenerate data. The two remaining degenerate sectors of the classical 3+1 dimensional theory are considered. First, with all densitized triad vectors…
In this work, resolutions will be given for commonly stated problems associated with a model that assumes that space and time are discretized (i.e., atomized). This model is in contrast to the continuous space-time model that is used in all…
We present a method that yields three decoupled covariant equations for three complex scalars, which completely govern electromagnetic perturbations of non-vacuum, locally rotationally symmetric class II spacetimes. One of these equations…
We propose a new discrete approximation to the Einstein equations, based on the Capovilla-Dell-Jacobson form of the action for the Ashtekar variables. This formulation is analogous to the Regge calculus in that the curvature has support on…
The application of numerical relativity to cosmological spacetimes is providing new insights into the behavior of Einstein's equations, beyond common approximations. In order for simulations to be performed as accurately and efficiently as…