The Well-posedness of the Null-Timelike Boundary Problem for Quasilinear Waves

The null-timelike initial-boundary value problem for a hyperbolic system of equations consists of the evolution of data given on an initial characteristic surface and on a timelike worldtube to produce a solution in the exterior of the worldtube. We establish the well-posedness of this problem for the evolution of a quasilinear scalar wave by means of energy estimates. The treatment is given in characteristic coordinates and thus provides a guide for developing stable finite difference algorithms. A new technique underlying the approach has potential application to other characteristic initial-boundary value problems.Comment: Version to appear in Class. Quantum Gra

The analysis and modeling of dilatational terms in compressible turbulence

It is shown that the dilatational terms that need to be modeled in compressible turbulence include not only the pressure-dilatation term but also another term - the compressible dissipation. The nature of these dilatational terms in homogeneous turbulence is explored by asymptotic analysis of the compressible Navier-Stokes equations. A non-dimensional parameter which characterizes some compressible effects in moderate Mach number, homogeneous turbulence is identified. Direct numerical simulations (DNS) of isotropic, compressible turbulence are performed, and their results are found to be in agreement with the theoretical analysis. A model for the compressible dissipation is proposed; the model is based on the asymptotic analysis and the direct numerical simulations. This model is calibrated with reference to the DNS results regarding the influence of compressibility on the decay rate of isotropic turbulence. An application of the proposed model to the compressible mixing layer has shown that the model is able to predict the dramatically reduced growth rate of the compressible mixing layer

The analysis and simulation of compressible turbulence

Compressible turbulent flows at low turbulent Mach numbers are considered. Contrary to the general belief that such flows are almost incompressible, (i.e., the divergence of the velocity field remains small for all times), it is shown that even if the divergence of the initial velocity field is negligibly small, it can grow rapidly on a non-dimensional time scale which is the inverse of the fluctuating Mach number. An asymptotic theory which enables one to obtain a description of the flow in terms of its divergence-free and vorticity-free components has been developed to solve the initial-value problem. As a result, the various types of low Mach number turbulent regimes have been classified with respect to the initial conditions. Formulae are derived that accurately predict the level of compressibility after the initial transients have disappeared. These results are verified by extensive direct numerical simulations of isotropic turbulence

Accurate black hole evolutions by fourth-order numerical relativity

We present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher order finite differencing for the BSSN formulation of Einstein's equations, but which is designed to be readily applicable to a broad class of formulations. We apply our techniques to a standard set of numerical relativity test problems, demonstrating the fourth-order accuracy of the solutions. Finally we apply our approach to binary black hole head-on collisions, calculating the waveforms of gravitational radiation generated and demonstrating significant improvements in waveform accuracy over second-order methods with typically achievable numerical resolution.Comment: 17 pages, 25 figure

Geometrization of metric boundary data for Einstein's equations

The principle part of Einstein equations in the harmonic gauge consists of a constrained system of 10 curved space wave equations for the components of the space-time metric. A well-posed initial boundary value problem based upon a new formulation of constraint-preserving boundary conditions of the Sommerfeld type has recently been established for such systems. In this paper these boundary conditions are recast in a geometric form. This serves as a first step toward their application to other metric formulations of Einstein's equations.Comment: Article to appear in Gen. Rel. Grav. volume in memory of Juergen Ehler

Exponential Decay for Small Non-Linear Perturbations of Expanding Flat Homogeneous Cosmologies

It is shown that during expanding phases of flat homogeneous cosmologies all small enough non-linear perturbations decay exponentially. This result holds for a large class of perfect fluid equations of state, but notably not for very ``stiff'' fluids as the pure radiation case

Initial boundary value problems for Einstein's field equations and geometric uniqueness

While there exist now formulations of initial boundary value problems for Einstein's field equations which are well posed and preserve constraints and gauge conditions, the question of geometric uniqueness remains unresolved. For two different approaches we discuss how this difficulty arises under general assumptions. So far it is not known whether it can be overcome without imposing conditions on the geometry of the boundary. We point out a natural and important class of initial boundary value problems which may offer possibilities to arrive at a fully covariant formulation.Comment: 19 page

Journal Staff

A time-dependent coordinate transformation of a constant coeffcient hyperbolic equation which results in a variable coeffcient problem is considered. By using the energy method, we derive well-posed boundary conditions for the continuous problem. It is shown that the number of boundary conditions depend on the coordinate transformation. By using Summation-by-Parts (SBP) operators for the space discretization and weak boundary conditions, an energy stable finite dieffrence scheme is obtained. We also show how to construct a time-dependent penalty formulation that automatically imposes the right number of boundary conditions. Numerical calculations corroborate the stability and accuracy of the approximations

Constraint and gauge shocks in one-dimensional numerical relativity

We study how different types of blow-ups can occur in systems of hyperbolic evolution equations of the type found in general relativity. In particular, we discuss two independent criteria that can be used to determine when such blow-ups can be expected. One criteria is related with the so-called geometric blow-up leading to gradient catastrophes, while the other is based upon the ODE-mechanism leading to blow-ups within finite time. We show how both mechanisms work in the case of a simple one-dimensional wave equation with a dynamic wave speed and sources, and later explore how those blow-ups can appear in one-dimensional numerical relativity. In the latter case we recover the well known ``gauge shocks'' associated with Bona-Masso type slicing conditions. However, a crucial result of this study has been the identification of a second family of blow-ups associated with the way in which the constraints have been used to construct a hyperbolic formulation. We call these blow-ups ``constraint shocks'' and show that they are formulation specific, and that choices can be made to eliminate them or at least make them less severe.Comment: 19 pages, 8 figures and 1 table, revised version including several amendments suggested by the refere

Temperature Modulates the Effects of Ocean Acidification on Intestinal Ion Transport in Atlantic Cod, Gadus morhua

CO2-driven seawater acidification has been demonstrated to enhance intestinal bicarbonate secretion rates in teleosts, leading to an increased release of CaCO3 under simulated ocean acidification scenarios. In this study, we investigated if increasing CO2 levels stimulate the intestinal acid–base regulatory machinery of Atlantic cod (Gadus morhua) and whether temperatures at the upper limit of thermal tolerance stimulate or counteract ion regulatory capacities. Juvenile G. morhua were acclimated for 4 weeks to three CO2 levels (550, 1200, and 2200 μatm) covering present and near-future natural variability, at optimum (10°C) and summer maximum temperature (18°C), respectively. Immunohistochemical analyses revealed the subcellular localization of ion transporters, including Na+/K+-ATPase (NKA), Na+/H+-exchanger 3 (NHE3), Na+/HCO−3 cotransporter (NBC1), pendrin-like Cl−/HCO−3 exchanger (SLC26a6), V-type H+-ATPase subunit a (VHA), and Cl− channel 3 (CLC3) in epithelial cells of the anterior intestine. At 10°C, proteins and mRNA were generally up-regulated for most transporters in the intestinal epithelium after acclimation to higher CO2 levels. This supports recent findings demonstrating increased intestinal HCO−3 secretion rates in response to CO2 induced seawater acidification. At 18°C, mRNA expression and protein concentrations of most ion transporters remained unchanged or were even decreased, suggesting thermal compensation. This response may be energetically favorable to retain blood HCO−3 levels to stabilize pHe, but may negatively affect intestinal salt and water resorption of marine teleosts in future oceans