Highly accurate numerical computation of implicitly defined volumes using the Laplace-Beltrami operator

This paper introduces a novel method for the efficient and accurate computation of the volume of a domain whose boundary is given by an orientable hypersurface which is implicitly given as the iso-contour of a sufficiently smooth level-set function. After spatial discretization, local approximation of the hypersurface and application of the Gaussian divergence theorem, the volume integrals are transformed to surface integrals. Application of the surface divergence theorem allows for a further reduction to line integrals which are advantageous for numerical quadrature. We discuss the theoretical foundations and provide details of the numerical algorithm. Finally, we present numerical results for convex and non-convex hypersurfaces embedded in cuboidal domains, showing both high accuracy and thrid- to fourth-order convergence in space.Comment: 25 pages, 17 figures, 3 table

A Simple Method for Computing Singular or Nearly Singular Integrals on Closed Surfaces

We present a simple, accurate method for computing singular or nearly singular integrals on a smooth, closed surface, such as layer potentials for harmonic functions evaluated at points on or near the surface. The integral is computed with a regularized kernel and corrections are added for regularization and discretization, which are found from analysis near the singular point. The surface integrals are computed from a new quadrature rule using surface points which project onto grid points in coordinate planes. The method does not require coordinate charts on the surface or special treatment of the singularity other than the corrections. The accuracy is about $O(h^3)$, where $h$ is the spacing in the background grid, uniformly with respect to the point of evaluation, on or near the surface. Improved accuracy is obtained for points on the surface. The treecode of Duan and Krasny for Ewald summation is used to perform sums. Numerical examples are presented with a variety of surfaces.Comment: to appear in Commun. Comput. Phy

Semiclassical S-matrix for black holes

We propose a semiclassical method to calculate S-matrix elements for two-stage gravitational transitions involving matter collapse into a black hole and evaporation of the latter. The method consistently incorporates back-reaction of the collapsing and emitted quanta on the metric. We illustrate the method in several toy models describing spherical self-gravitating shells in asymptotically flat and AdS space-times. We find that electrically neutral shells reflect via the above collapse-evaporation process with probability exp(-B), where B is the Bekenstein-Hawking entropy of the intermediate black hole. This is consistent with interpretation of exp(B) as the number of black hole states. The same expression for the probability is obtained in the case of charged shells if one takes into account instability of the Cauchy horizon of the intermediate Reissner-Nordstrom black hole. Our semiclassical method opens a new systematic approach to the gravitational S-matrix in the non-perturbative regime.Comment: 41 pages, 13 figures; Introduction rewritten, references added; journal versio

An unfitted discontinuous Galerkin scheme for conservation laws on evolving surfaces

Motivated by considering partial differential equations arising from conservation laws posed on evolving surfaces, a new numerical method for an advection problem is developed and simple numerical tests are performed. The method is based on an unfitted discontinuous Galerkin approach where the surface is not explicitly tracked by the mesh which means the method is extremely flexible with respect to geometry. Furthermore, the discontinuous Galerkin approach is well-suited to capture the advection driven by the evolution of the surface without the need for a space-time formulation, back-tracking trajectories or streamline diffusion. The method is illustrated by a one-dimensional example and numerical results are presented that show good convergence properties for a simple test problem

Open inflation from non-singular instantons: Wrapping the universe with a membrane

The four-form field recently considered by Hawking and Turok couples naturally to a charged membrane, across which the effective cosmological constant has a discontinuity. We present instantons for the creation of an open inflationary universe surrounded by a membrane. They can also be used to describe the nucleation of a membrane on a pre-existing inflationary background. This process typically decreases the value of the effective cosmological constant and may lead to a novel scenario of eternal inflation. Moreover, by coupling the inflaton field to the membrane, the troublesome singularities which arise in the Hawking-Turok model can be eliminated.Comment: 23 pages, LaTeX2e, 4 figure