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Well-posedness of the initial (boundary) value problem is an essential property, both of meaningful physical models and of numerical applications. To prove well-posedness of wave-type equations their level of hyperbolicity is an essential…
Original abstract: "We construct periodic solutions of nonlinear wave equations using analytic continuation. The construction applies in particular to Einstein equations, leading to infinite-dimensional families of time-periodic solutions…
The importance and usefulness of renormalization are emphasized in nonrelativistic quantum mechanics. The momentum space treatment of both two-body bound state and scattering problems involving some potentials singular at the origin…
A generalized geometric method is developed for constructing exact solutions of gravitational field equations in Einstein theory and generalizations. First, we apply the formalism of nonholonomic frame deformations (formally considered for…
In many time-harmonic electromagnetic wave problems, the considered geometry exhibits an axial symmetry. In this case, by exploiting a Fourier expansion along the azimuthal direction, fully three-dimensional (3D) calculations can be carried…
We linearize the Einstein equations when the metric is Bondi-Sachs, when the background is Schwarzschild or Minkowski, and when there is a matter source in the form of a thin shell whose density varies with time and angular position. By…
Results from helically symmetric scalar field models and first results from a convergent helically symmetric binary neutron star code are reported here; these are models stationary in the rotating frame of a source with constant angular…
We design a primal-dual stabilized finite element method for the numerical approximation of a data assimilation problem subject to the acoustic wave equation. For the forward problem, piecewise affine, continuous, finite element functions…
In this paper we discuss a new approach to the quasinormal-mode problem in general relativity. By combining a characteristic formulation of the perturbation equations with the integration of a suitable phase-function for a complex valued…
Numerical relativity is an essential tool for solving Einstein's equations of general relativity for dynamical systems characterized by high velocities and strong gravitational fields. The implementation of new algorithms that can solve…
It is well known, thanks to Lax-Wendroff theorem, that the local conservation of a numerical scheme for a conservative hyperbolic system is a simple and systematic way to guarantee that, if stable, a scheme will provide a sequence of…
I discuss the conformal approach to the numerical simulation of radiating isolated systems in general relativity. The method is based on conformal compactification and a reformulation of the Einstein equations in terms of rescaled…
We present numerical solutions of several spacetimes of physical interest, including binary black hole mergers, in shift-symmetric Einstein-scalar-Gauss-Bonnet (ESGB) gravity, and describe our methods for solving the full equations of…
We compute families of solutions to the Einstein vacuum equations of the type of Brill waves, but with slow fall-off towards spatial infinity. We prove existence and uniqueness of solutions for physical data and numerically construct some…
We consider the scalar wave equation with power nonlinearity in n+1 dimensions. Unlike most previous numerical studies, we go beyond the radial case and do not assume any symmetries for n=3, and we only impose an SO(n-1) symmetry in higher…
The spectral triple approach to noncommutative geometry allows one to develop the entire standard model (and supersymmetric extensions) of particle physics from a purely geometry stand point and thus treats both gravity and particle physics…
This is a study of the Euler equations for free surface water waves in the case of varying bathymetry, considering the problem in the shallow water scaling regime. In the case of rapidly varying periodic bottom boundaries this is a problem…
An unsteady problem is considered for a space-fractional equation in a bounded domain. A first-order evolutionary equation involves the square root of an elliptic operator of second order. Finite element approximation in space is employed.…
In this paper, we prove the first asymptotic completeness result for a scalar quasilinear wave equation satisfying the weak null condition. The main tool we use in the study of this equation is the geometric reduced system introduced in…
This paper concerns the numerical resolution of a data completion problem for the time-harmonic Maxwell equations in the electric field. The aim is to recover the missing data on the inaccessible part of the boundary of a bounded domain…