Search results for "RIEMANN"
showing 10 items of 254 documents
A Numerical Study of Relativistic Jets
1996
Numerical simulations of supersonic jets are able to explain the structures observed in many VLA images of radio sources. The improvements achieved in classical simulations (see Hardee, these proceedings) are in contrast with the almost complete lack of relativistic simulations the reason being that numerical difficulties arise from the highly relativistic flows typical of extragalactic jets. For our study, we have developed a two-dimensional code which is based on (i) an explicit conservative differencing of the special relativistic hydrodynamics (SRH) equations and (ii) the use of an approximate Riemann solver (see Marti et al. 1995a,b and references therein).
On the Measurements of Numerical Viscosity and Resistivity in Eulerian MHD Codes
2016
We propose a simple ansatz for estimating the value of the numerical resistivity and the numerical viscosity of any Eulerian MHD code. We test this ansatz with the help of simulations of the propagation of (magneto)sonic waves, Alfven waves, and the tearing mode instability using the MHD code Aenus. By comparing the simu- lation results with analytical solutions of the resistive-viscous MHD equations and an empirical ansatz for the growth rate of tearing modes we measure the numerical viscosity and resistivity of Aenus. The comparison shows that the fast-magnetosonic speed and wavelength are the characteristic velocity and length, respectively, of the aforementioned (relatively simple) syst…
On Riemann’s Ideas on Space and Schwinger’s Treatment of Low-Energy Pion-Nucleon Physics
2020
We begin with a demonstration of how great an influence Riemann’s habilitation essay had on the development of field theory. His ideas about the origin of physical space and the importance of a metric field were clearly outlined as early as 1854, and praised highly by the old C.F. Gauss, who died 1 year later. There is but one formula in Riemann’s article. This formula and its relevance will be explained at the beginning of the present chapter. The basic principle which is omnipresent in Riemann’s entire work is to understand the physical behavior of nature from its smallness. Hence partial differential equations stand at the beginning of any field theory. In our case, it is not the metric …
3D Relativistic Hydrodynamics
2007
We review the evolution of the numerical techniques applied in relativistic hydrodynamics since the sixties until today. We focus our attention on the state-of- the-art high-resolution shock-capturing methods and the astrophysical applications involving three-dimensional simulations.
Generalized Buckley–Leverett theory for two-phase flow in porous media
2011
Hysteresis and fluid entrapment pose unresolved problems for the theory of flow in porous media. A generalized macroscopic mixture theory for immiscible two-phase displacement in porous media (Hilfer 2006b Phys. Rev. E 73 016307) has introduced percolating and nonpercolating phases. It is studied here in an analytically tractable hyperbolic limit. In this limit a fractional flow formulation exists, that resembles the traditional theory. The Riemann problem is solved analytically in one dimension by the method of characteristics. Initial and boundary value problems exhibit shocks and rarefaction waves similar to the traditional Buckley-Leverett theory. However, contrary to the traditional th…
Numerical 3+1 general relativistic magnetohydrodynamics: a local characteristic approach
2005
We present a general procedure to solve numerically the general relativistic magnetohydrodynamics (GRMHD) equations within the framework of the 3+1 formalism. The work reported here extends our previous investigation in general relativistic hydrodynamics (Banyuls et al. 1997) where magnetic fields were not considered. The GRMHD equations are written in conservative form to exploit their hyperbolic character in the solution procedure. All theoretical ingredients necessary to build up high-resolution shock-capturing schemes based on the solution of local Riemann problems (i.e. Godunov-type schemes) are described. In particular, we use a renormalized set of regular eigenvectors of the flux Jac…
Relativistic Magnetohydrodynamics: Renormalized eigenvectors and full wave decomposition Riemann solver
2010
We obtain renormalized sets of right and left eigenvectors of the flux vector Jacobians of the relativistic MHD equations, which are regular and span a complete basis in any physical state including degenerate ones. The renormalization procedure relies on the characterization of the degeneracy types in terms of the normal and tangential components of the magnetic field to the wavefront in the fluid rest frame. Proper expressions of the renormalized eigenvectors in conserved variables are obtained through the corresponding matrix transformations. Our work completes previous analysis that present different sets of right eigenvectors for non-degenerate and degenerate states, and can be seen as…
An introduction to relativistic hydrodynamics
2007
We review formulations of the equations of (inviscid) general relativistic hydrodynamics and (ideal) magnetohydrodynamics, along with methods for their numerical solution. Both systems can be cast as first-order, hyperbolic systems of conservation laws, following the explicit choice of an Eulerian observer and suitable fluid and magnetic field variables. During the last fifteen years, the so-called (upwind) high-resolution shock-capturing schemes based on Riemann solvers have been successfully extended from classical to relativistic fluid dynamics, both special and general. Nowadays, general relativistic hydrodynamical simulations in relativistic astrophysics are routinely performed, partic…
Non-Riemannian geometry: towards new avenues for the physics of modified gravity
2015
Less explored than their metric (Riemannian) counterparts, metric-affine (or Palatini) theories bring an unexpected phenomenology for gravitational physics beyond General Relativity. Lessons of crystalline structures, where the presence of defects in their microstructure requires the use of non-Riemannian geometry for the proper description of their properties in the macroscopic continuum level, are discussed. In this analogy, concepts such as wormholes and geons play a fundamental role. Applications of the metric-affine formalism developed by the authors in the last three years are reviewed.
A Remark on an Overdetermined Problem in Riemannian Geometry
2016
Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let \(\varOmega \subset M\) be a bounded domain, with \(O \in \varOmega \), and consider the problem \(\varDelta _p u = -1\ \mathrm{in}\ \varOmega \) with \(u=0\ \mathrm{on}\ \partial \varOmega \), where \(\varDelta _p\) is the p-Laplacian of g. We prove that if the normal derivative \(\partial _{\nu }u\) of u along the boundary of \(\varOmega \) is a function of d satisfying suitable conditions, then \(\varOmega \) must be a geodesic ball. In particular, our result applies to open balls of \(\mathbb {R}^n\) equipped with a rotationally symmetric metr…