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We describe the Einstein Toolkit, a community-driven, freely accessible computational infrastructure intended for use in numerical relativity, relativistic astrophysics, and other applications. The Toolkit, developed by a collaboration…
This paper introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second order partial differential equations. Based on a discrete…
This paper is to introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary…
In the harmonic description of general relativity, the principle part of Einstein's equations reduces to 10 curved space wave equations for the componenets of the space-time metric. We present theorems regarding the stability of several…
We present SphericalNR, a new framework for the publicly available Einstein Toolkit that numerically solves the Einstein field equations coupled to the equations of general relativistic magnetohydrodynamics (GRMHD) in a 3+1 split of…
Numerical studies of the dynamics of gravitational systems, e.g., black hole-neutron star systems, require physical and constraint-satisfying initial data. In this article, we present the newly developed pseudo-spectral code Elliptica, an…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
GRTresna is a multigrid solver designed to solve the constraint equations for the initial data required in numerical relativity simulations. In particular, it is focussed on scenarios with fundamental fields around black holes and…
We consider geometric multigrid methods for the solution of linear systems arising from isogeometric discretizations of elliptic partial differential equations. For classical finite elements, such methods are well known to be fast solvers…
A considerable amount of attention has been given to discontinuous Galerkin methods for hyperbolic problems in numerical relativity, showing potential advantages of the methods in dealing with hydrodynamical shocks and other…
We apply physics-informed neural networks (PINNs) to first-order two-scale periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp…
The search for solutions of Einstein's equations representing relativistic cosmological models with a discrete matter content has been remarkably fruitful in the last decade. In this review we discuss the progress made in the study of a…
We use rigorous techniques from numerical analysis of hyperbolic equations in bounded domains to construct stable finite-difference schemes for Numerical Relativity, in particular for their use in black hole excision. As an application, we…
This work presents a novel methodology for deriving stationary and axially symmetric solutions to Einstein field equations using the 1+3 tetrad formalism. This approach reformulates the Einstein equations into first order scalar equations,…
We propose some multigrid methods for solving the algebraic systems resulting from finite element approximations of space fractional partial differential equations (SFPDEs). It is shown that our multigrid methods are optimal, which means…
Apparent horizon plays an important role in numerical relativity as it provides a tool to characterize the existence and properties of black holes on three-dimensional spatial slices in 3+1 numerical spacetimes. Apparent horizon finders…
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…
We study a boundary-value quasilinear elliptic problem on a generic time scale. Making use of the fixed-point index theory, sufficient conditions are given to obtain existence, multiplicity, and infinite solvability of positive solutions.
A method is introduced for solving Einstein's equations using two distinct coordinate systems. The coordinate basis vectors associated with one system are used to project out components of the metric and other fields, in analogy with the…
Outer boundary conditions for strongly and symmetric hyperbolic formulations of 3D Einstein's field equations with a live gauge condition are discussed. The boundary conditions have the property that they ensure constraint propagation and…