Simulation of Asymptotically AdS5 Spacetimes with a Generalized Harmonic Evolution Scheme

Motivated by the gauge/gravity duality, we introduce a numerical scheme based on generalized harmonic evolution to solve the Einstein field equations on asymptotically anti-de Sitter (AdS) spacetimes. We work in global AdS5, which can be described by the (t,r,\chi,\theta,\phi) spherical coordinates adapted to the R{\times}S3 boundary. We focus on solutions that preserve an SO(3) symmetry that acts to rotate the 2-spheres parametrized by \theta,\phi. In the boundary conformal field theory (CFT), the way in which this symmetry manifests itself hinges on the way we choose to embed Minkowski space in R{\times}S3. We present results from an ongoing study of prompt black hole formation via scalar field collapse, and explore the subsequent quasi-normal ringdown. Beginning with initial data characterized by highly distorted apparent horizon geometries, the metrics quickly evolve, via quasi-normal ringdown, to equilibrium static black hole solutions at late times. The lowest angular number quasi-normal modes are consistent with the linear modes previously found in perturbative studies, whereas the higher angular modes are a combination of linear modes and of harmonics arising from non-linear mode-coupling. We extract the stress energy tensor of the dual CFT on the boundary, and find that despite being highly inhomogeneous initially, it nevertheless evolves from the outset in a manner that is consistent with a thermalized N=4 SYM fluid. As a first step towards closer contact with relativistic heavy ion collision physics, we map this solution to a Minkowski piece of the R{\times}S3 boundary, and obtain a corresponding fluid flow in Minkowski space

Hydrodynamic scaling limit of continuum solid-on-solid model

A fourth-order nonlinear evolution equation is derived from a microscopic model for surface diffusion, namely, the continuum solid-on-solid model. We use the method developed by Varadhan for the computation of hydrodynamic scaling limit of nongradient models. What distinguishes our model from other models discussed so far is the presence of two conservation laws for the dynamics in a nonperiodic box and the complex dynamics that is not nearest-neighbor. Along the way, a few steps has to be adapted to our new context. As a byproduct of our main result we also derive the hydrodynamic scaling limit of a perturbation of continuum solid-on-solid model, a model that incorporates both surface diffusion and surface electromigration

Asymptotic profiles for the Cauchy problem of damped beam equation with two variable coefficients and derivative nonlinearity

In this article we investigate the asymptotic profile of solutions for the Cauchy problem of the nonlinear damped beam equation with two variable coefficients: $$\partial_t^2 u + b(t) \partial_t u - a(t) \partial_x^2 u + \partial_x^4 u = \partial_x \left( N(\partial_x u) \right).$$ In the authors' previous article [17], the asymptotic profile of solutions for linearized problem ($N \equiv 0$) was classified depending on the assumptions for the coefficients $a(t)$ and $b(t)$ and proved the asymptotic behavior in effective damping cases. We here give the conditions of the coefficients and the nonlinear term in order that the solution behaves as the solution for the heat equation: $b(t) \partial_t u - a(t) \partial_x^2 u=0$ asymptotically as $t \to \infty$.Comment: 32 pages, 1 figur

StaRMAP - A second order staggered grid method for spherical harmonics moment equations of radiative transfer

We present a simple method to solve spherical harmonics moment systems, such as the the time-dependent $P_N$ and $SP_N$ equations, of radiative transfer. The method, which works for arbitrary moment order $N$, makes use of the specific coupling between the moments in the $P_N$ equations. This coupling naturally induces staggered grids in space and time, which in turn give rise to a canonical, second-order accurate finite difference scheme. While the scheme does not possess TVD or realizability limiters, its simplicity allows for a very efficient implementation in Matlab. We present several test cases, some of which demonstrate that the code solves problems with ten million degrees of freedom in space, angle, and time within a few seconds. The code for the numerical scheme, called StaRMAP (Staggered grid Radiation Moment Approximation), along with files for all presented test cases, can be downloaded so that all results can be reproduced by the reader.Comment: 28 pages, 7 figures; StaRMAP code available at http://www.math.temple.edu/~seibold/research/starma