We present a numerical method to solve the equations of general relativistic hydrodynamics in a given external gravitational field. The method is based on a generalization of Roe’s approximate Riemann solver for the non relativistic Euler equations in Cartesian coordinates. The new method is applied to a set of standard test problems for general relativistic hydrodynamics, and is shown to perfo...

The purpose of this paper is to describe geometrically discrete Lagrangian and Hamiltonian Mechanics on Lie groupoids. From a variational principle we derive the discrete Euler-Lagrange equations and we introduce a symplectic 2-section, which is preserved by the Lagrange evolution operator. In terms of the discrete Legendre transformations we define the Hamiltonian evolution operator which is a...

Abstract The Lagrangian average (LA) of the ideal fluid equations preserves their fundamental transport structure. This transport structure is responsible for the Kelvin circulation theorem of the LA flow and, hence, for its potential vorticity convection and helicity conservation. We show that Lagrangian averaging also preserves the Euler–Poincaré variational framework that implies the exact i...

The Euler equations for water waves in any depth have been shown to have solitary wave solutions when the effect of surface tension is included. This paper proposes three quadratic model equations for these types of waves in infinite depth with a two-dimensional fluid domain. One model is derived directly from the Euler equations. Two further simpler models are proposed, both having the full gr...

Non blow-up of the 3D incompressible Euler Equations is proven for a class of threedimensional initial data characterized by uniformly large vorticity in bounded cylindrical domains. There are no conditional assumptions on the properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast singu...

In [C2], Chemin shows that solutions of the Navier-Stokes equations in R for an incompressible fluid whose initial vorticity lies in L ∩ L∞ converge in the zero-viscosity limit in the L–norm to a solution of the Euler equations, convergence being uniform over any finite time interval. In [Y2], Yudovich assumes an initial vorticity lying in L for all p ≥ p0, and establishes the uniqueness of sol...

Multiple numerical solutions of inviscid transonic flow over several airfoils exist at certain Mach number and angle of attack. Global linear stability analysis of the multiple solutions is conducted in this paper. Linear perturbation equations of the Euler equations around a steady-state solution are formed, an eigenvalue problem is then constructed using the modal analysis approach, then it i...

Recently, Berenger introduced a Perfectly Matched Layer (PML) technique for absorbing electromagnetic waves. In the present paper, a perfectly matched layer is proposed for absorbing out-going two-dimensional waves in a uniform mean ow, governed by linearized Euler equations. It is well known that the linearized Euler equations support acoustic waves, which travel with the speed of sound relati...