Some implications of signature-change in cosmological models of loop quantum gravity

Signature change at high density has been obtained as a possible consequence of deformed space-time structures in models of loop quantum gravity. This article provides a conceptual discussion of implications for cosmological scenarios, based on an application of mathematical results for mixed-type partial differential equations (the Tricomi problem). While the effective equations from which signature change has been derived are shown to be locally regular and therefore reliable, the underlying theory of loop quantum gravity may face several global problems in its semiclassical solutions.Comment: 35 pages, 5 figure

Causality of fluid dynamics for high-energy nuclear collisions

Dissipative relativistic fluid dynamics is not always causal and can favor superluminal signal propagation under certain circumstances. On the other hand, high-energy nuclear collisions have a microscopic description in terms of QCD and are expected to follow the causality principle of special relativity. We discuss under which conditions the fluid evolutions for a radial expansion are hyperbolic and how the properties of the solutions are encoded in the associated characteristic curves. The expansion dynamics is causal in relativistic sense if the characteristic velocities are smaller than the speed of light. We obtain a concrete inequality from this constraint and discuss how it can be violated for certain initial conditions. We argue that causality poses a bound to the applicability of relativistic fluid dynamics. }Comment: 23 pages, 13 figures; Added references, corrected typos, added discussion as section 2, results unchange

Hyperboloidal layers for hyperbolic equations on unbounded domains

We show how to solve hyperbolic equations numerically on unbounded domains by compactification, thereby avoiding the introduction of an artificial outer boundary. The essential ingredient is a suitable transformation of the time coordinate in combination with spatial compactification. We construct a new layer method based on this idea, called the hyperboloidal layer. The method is demonstrated on numerical tests including the one dimensional Maxwell equations using finite differences and the three dimensional wave equation with and without nonlinear source terms using spectral techniques.Comment: 23 pages, 23 figure

Uncertainty Quantification for Linear Hyperbolic Equations with Stochastic Process or Random Field Coefficients

In this paper hyperbolic partial differential equations with random coefficients are discussed. Such random partial differential equations appear for instance in traffic flow problems as well as in many physical processes in random media. Two types of models are presented: The first has a time-dependent coefficient modeled by the Ornstein--Uhlenbeck process. The second has a random field coefficient with a given covariance in space. For the former a formula for the exact solution in terms of moments is derived. In both cases stable numerical schemes are introduced to solve these random partial differential equations. Simulation results including convergence studies conclude the theoretical findings

Wave Solutions

In classical continuum physics, a wave is a mechanical disturbance. Whether the disturbance is stationary or traveling and whether it is caused by the motion of atoms and molecules or the vibration of a lattice structure, a wave can be understood as a specific type of solution of an appropriate mathematical equation modeling the underlying physics. Typical models consist of partial differential equations that exhibit certain general properties, e.g., hyperbolicity. This, in turn, leads to the possibility of wave solutions. Various analytical techniques (integral transforms, complex variables, reduction to ordinary differential equations, etc.) are available to find wave solutions of linear partial differential equations. Furthermore, linear hyperbolic equations with higher-order derivatives provide the mathematical underpinning of the phenomenon of dispersion, i.e., the dependence of a wave's phase speed on its wavenumber. For systems of nonlinear first-order hyperbolic equations, there also exists a general theory for finding wave solutions. In addition, nonlinear parabolic partial differential equations are sometimes said to posses wave solutions, though they lack hyperbolicity, because it may be possible to find solutions that translate in space with time. Unfortunately, an all-encompassing methodology for solution of partial differential equations with any possible combination of nonlinearities does not exist. Thus, nonlinear wave solutions must be sought on a case-by-case basis depending on the governing equation.Comment: 22 pages, 3 figures; to appear in the Mathematical Preliminaries and Methods section of the Encyclopedia of Thermal Stresses, ed. R.B. Hetnarski, Springer (2014), to appea

The Initial-Boundary Value Problem in General Relativity

In this article we summarize what is known about the initial-boundary value problem for general relativity and discuss present problems related to it.Comment: 11 pages, 2 figures. Contribution to a special volume for Mario Castagnino's seventy fifth birthda

Finite element approximation of high-dimensional transport-dominated diffusion problems

High-dimensional partial differential equations with nonnegative characteristic form arise in numerous mathematical models in science. In problems of this kind, the computational challenge of beating the exponential growth of complexity as a function of dimension is exacerbated by the fact that the problem may be transport-dominated. We develop the analysis of stabilised sparse finite element methods for such high-dimensional, non-self-adjoint and possibly degenerate partial differential equations.\ud \ud (Presented as an invited lecture under the title "Computational multiscale modelling: Fokker-Planck equations and their numerical analysis" at the Foundations of Computational Mathematics conference in Santander, Spain, 30 June - 9 July, 2005.