Related papers: Gmunu: Toward multigrid based Einstein field equat…

We present a unified hard-constraint framework for solving geometrically complex PDEs with neural networks, where the most commonly used Dirichlet, Neumann, and Robin boundary conditions (BCs) are considered. Specifically, we first…

The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear…

This work is focused in the study of analytic anisotropic solutions to Einstein's field equations, describing spherically symmetric and static configurations by way of the gravitational decoupling through the method of Minimal Geometric…

We investigate solutions of the classical Einstein or supergravity equations that solve any set of quantum corrected Einstein equations in which the Einstein tensor plus a multiple of the metric is equated to a symmetric conserved tensor…

We report on recent extensions and improvements to the Legolas code, which is an open-source, finite element-based numerical framework to solve the linearised (magneto)hydrodynamic equations for a three-dimensional force- and thermally…

In this paper we use the GeneralizedMultiscale Finite ElementMethod (GMsFEM) framework, introduced in [20], in order to solve nonlinear elliptic equations with high-contrast coefficients. The proposed solution method involves linearizing…

Recent works by the second author and Maxwell et al. have shown that the Einstein-scalar field conformal constraint equations are highly complex and generally intractable, even in the vacuum case. In this article, to gain a clearer…

In this paper we propose an extension of the generalized Lagrangian multiplier method (GLM) of Munz et al. (JCP 2000, JCP 2002), which was originally conceived for the numerical solution of the Maxwell and MHD equations with divergence-type…

We introduce an additive Gaussian process framework accounting for monotonicity constraints and scalable to high dimensions. Our contributions are threefold. First, we show that our framework enables to satisfy the constraints everywhere in…

Implicit methods and GPU parallelization are two distinct yet powerful strategies for accelerating high-order CFD algorithms. However, few studies have successfully integrated both approaches within high-speed flow solvers. The core…

In this work, we propose a robust and easily implemented algebraic multigrid method as a stand-alone solver or a preconditioner in Krylov subspace methods for solving either symmetric and positive definite or saddle point linear systems of…

In numerical relativity simulations with non-trivial matter configurations, one must solve the Hamiltonian and momentum constraints of the ADM formulation for the metric variables in the initial data. We introduce a new scheme based on the…

We consider a standard elliptic partial differential equation and propose a geometric multigrid algorithm based on Dirichlet-to-Neumann (DtN) maps for hybridized high-order finite element methods. The proposed unified approach is applicable…

In this manuscript, we present a collective multigrid algorithm to solve efficiently the large saddle-point systems of equations that typically arise in PDE-constrained optimization under uncertainty, and develop a novel convergence…

We present a new approach for stably evolving general relativistic magnetohydrodynamic (GRMHD) simulations in regions where the magnetization $\sigma=b^2/\rho c^2$ becomes large. GRMHD codes typically struggle to evolve plasma above…

Writing high performance solvers for engineering applications is a delicate task. These codes are often developed on an application to application basis, highly optimized to solve a certain problem. Here, we present our work on developing a…

We introduce our newly developed two different, three dimensional magneto hydrodynamical codes in detail. One of our codes is written in the Newtonian limit (NMHD) and the other is in the fully general relativistic code (GRMHD). Both codes…

Due to the nonseparability of the post-Newtonian (PN) Hamiltonian systems of compact objects, the symplectic methods that admit the linear error growth and the near preservation of first integrals are always implicit as explicit symplectic…

We assemble the equations of general relativistic magnetohydrodynamics (MHD) in 3+1 form. These consist of the complete coupled set of Maxwell equations for the electromagnetic field, Einstein's equations for the gravitational field, and…

This is the second in a series of papers on the construction and validation of a three-dimensional code for the solution of the coupled system of the Einstein equations and of the general relativistic hydrodynamic equations, and on the…