Abstracts

Reparametrization invariance, constraints and perfect actions in Regge calculus

Reparametrization invariance is one of the key aspects in General Relativity, as it incorporates the principle of covariance and background-independence. It is manifest as diffeomorphism symmetry of the theory. In this talk it is described in which sense this diffeomorphism symmetry is broken in 4D Regge Calculus by the introduction of a triangulation, and how this poses problems for e.g. identifying the physical degrees of freedom. Furthermore, the perfect action approach is presented, which aims at solving this problem, as well as its connections to finding an implementation of the constraints in a discrete Hamiltonian formulation of Regge Calculus. We also present the example of 3D Regge Calculus with cosmological constant, where the problem also exists and can be solved completely by the perfect action method, and comment on the status of the program in 4D.

Bruno Benedetti (TU Berlin)

Locally constructible manifolds

Durhuus and Jonsson (1995) introduced the class of locally constructible (LC) spheres and showed that there are only exponentially many combinatorial types of LC simplicial 3-spheres. Such upper bounds are crucial for the convergence of the dynamical triangulations model for 3D quantum gravity. We characterize the LC property for manifolds and show that plenty of non-LC d-spheres exist for each d>2. At the same time, we show how discrete Morse theory yields exponential cutoffs of the class of all triangulations of manifolds with N facets.

Snørre H. Christiansen (U Oslo)

	On the linearization of Regge calculus in 3D
	We relate linearization of Regge calculus in 3D to a discrete elasticity complex. We show that the eigenvalue problem for the Saint-Venant operator converges when discretized by Regge metrics.

Jörg Frauendiener (U Otago)

Discrete geometric structures in general relativity

The formalism of discrete differential forms has been used very successfully in computational electrodynamics. It is based on the idea that only the observables (i.e., the electromagnetic field) should be discretised and that coordinates should not possess any relevance in the numerical method. This is reflected in the fact that Maxwell's theory can be written entirely in geometric terms using differential forms. Einstein's theory is entirely geometric as well and can also be written in terms of differential forms. In this talk I will describe an attempt to discretise Einstein's theory in a way similar to Maxwell's theory. I will describe the advantages and point out disadvantages. I will conclude with some remarks about more general discrete structures on manifolds.

Joachim Giesen (U Jena) (work done with Madhusudan Manjunath)

Minimizing absolute Gaussian curvature locally

One of the remaining challenges when reconstructing a surface from a finite sample is recovering non-smooth surface features like sharp edges. There is practical evidence showing that a two step approach could be an aid to this problem, namely, first computing a polyhedral reconstruction isotopic to the sampled surface, and secondly minimizing the absolute Gaussian curvature of this reconstruction globally. The first step ensures topological correctness and the second step improves the geometric accuracy of the reconstruction in the presence of sharp features without changing its topology. Unfortunately it is computationally hard to minimize the absolute Gaussian curvature globally. Hence, I will discuss a local variant of absolute Gaussian curvature minimization problem which is still meaningful in the context of surface fairing. Absolute Gaussian curvature like Gaussian curvature is concentrated at the vertices of a polyhedral surface embedded into R^3. Local optimization tries to move a single vertex in space such that the absolute Gaussian curvature at this vertex is minimized.

Michael Holst (UC San Diego)

Analysis and Finite Element Approximation of the Einstein Constraints

In the first part of this lecture, we examine the geometric elliptic system forming the constraints in the Einstein equations, and consider a thirty-five-year-old open question involving existence of solutions to the constraint equations on space-like hyper-surfaces with arbitrarily prescribed mean extrinsic curvature. We give the first partial answer to this question using a priori estimates and a new type of topological fixed-point argument. In the second part of the lecture, we develop two convergence frameworks for abstract adaptive finite element (AFEM) algorithms for nonlinear elliptic equations on Riemannian manifolds, and then use the a priori estimates from the first part of the lecture to obtain some AFEM convergence results for the Einstein constraints. In the last part of the lecture, we give a new analysis of surface finite element methods for problems such as the Einstein constraints, by first analyzing variational crimes in Hilbert complexes, and then applying the results to finite element exterior calculus on hypersurfaces.

Waner A. Miller (FAU)

Regge Calculus as a Dynamical Theory: The Fundamental Role of Dual Tessellations

We emphasize the fundamental role that the circumcentric dual tessellation plays in the development of Regge calculus (RC) and in our understanding the dynamical degrees of freedom of the discrete gravitational field. Cartan's moment-of-rotation construction provides the best route we are aware of to gain an understanding RC as a dynamical theory, including the diffeomorphic degrees of freedom. After we introduce the curvature of unstructured meshes, we will derive the RC equations, Regge-Hilbert action and Bianchi identities. In the second part of the talk, we introduce the thin-sandwich initial-value procedure for RC. Finally, we will outline some of our group's numerical applications of RC to geometrodynamics.

Ilia Musco (U Oslo)

Numerical aspects of primordial black hole formation

Critical collapse is a well-known problem in numerical relativity and it has been shown to play a key role in the formation of primordial black holes when using the model of a single perfect fluid to describe the collapse of growing-mode cosmological perturbations in the early universe. We have made a detailed study of this problem using a spherically symmetric Lagrangian code with null slicing: during the hydrodynamical evolution we have observed formation of semi-void regions with very low densities traversed by relativistic winds with extremely high outward velocities, as well as the extremely dense regions where the black hole is being produced. Obtaining a satisfactory numerical solution of the Einstein + hydro equations in this context is a challenging problem that we have tackled using a purpose-built adaptive mesh refinement scheme which adds and subtracts grid zones so as to maintain adequate resolution. This has been used successfully with more than thirty levels of refinement, giving a zone-width variation by factors of up to 10^9. Using this, we have been able to demonstrate a scaling relation consistent with the theory of critical collapse for a perfect fluid, extending down to machine round-off. The collapsing region is characterized by nearly self-similar behaviour, and no shock waves are seen to arise in this context.

Frank Peuker (U Jena)

Simulations of testbed space-times in Regge Calculus: Least-squares solutions, convergence and application of external grid generators

In this talk, different aspects of the numerical implementation of time-evolutions in Regge Calculus are investigated. It is shown that for almost flat space-times, a least-squares solution of the overdetermined system of equations (QR scheme) is superior to the utilization of a simplicial Bianchi identity. By means of this QR scheme we investigate convergence orders in space and time by applying standard integral norms to the metric components. A Courant condition is revealed which must hold for stable time-evolutions. Furthermore, we apply the QR scheme to a ball-shaped domain meshed by a free external grid generator to evolve Schwarzschild space-time.

David Radice (AEI)

Very high order discontinuous Galerkin methods for general relativistic hydrodynamics in spherical symmetry

Runge-Kutta discontinuous Galerkin (RKDG) methods are arbitrary high resolution methods which combine characteristics of spectral and finite volume methods, allowing for very high accuracy up to the shock fronts and for sharp shock capturing. These methods have been particularly successful in Newtonian computational fluid dynamics, thanks to their nearly-linear scalability on unstructured multi-domain meshes. We present a proof-of-concept 1D, RKDG, general relativistic hydrodynamical code employing a Galerkin-Legendre spectral collocation of the equations and a spectral filtering stabilization technique. We discuss the properties of the proposed scheme and show the results obtained in some common test like shock tubes and spherical accretion into a black hole.

Ronny Richter (U Tübingen)

	On the application of discrete differential forms in general relativity
	To investigate the applicability of discrete differential forms in numerical relativity the method was implemented for spherically symmetric and cosmological space times. In this talk results of the simulations are presented and some details of the implementation are discussed.

Olindo Zanotti (AEI)

Resistive relativistic magnetohydrodynamics with Galerkin methods

I present the first better than second order accurate method in space and time for the numerical solution of the resistive relativistic magnetohydrodynamics (RRMHD) equations on unstructured meshes in multiple space dimensions. The nonlinear system under consideration is purely hyperbolic and contains a source term, that for the evolution of the electric field, that becomes stiff for low values of the resistivity. For the spatial discretization a high order P_N P_M scheme is used while a high order accurate unsplit time discretization is achieved using an element-local space-time discontinuous Galerkin approach. The divergence free character of the magnetic field is accounted for through the divergence cleaning procedure. The proposed method can handle equally well the resistive regime and the stiff limit of ideal relativistic MHD. For these reasons it provides a powerful tool for relativistic astrophysical simulations involving the appearance of magnetic reconnection.

Gerhard Zumbusch (U Jena)

Finite elements methods for wave equations

Numerical schemes for second order wave equations are developed. Space-time is discretized with continuous and discontinuous Galerkin methods, i.e. Finite Element methods and symmetric/non-symmetric interior penalty discontinuous Galerkin methods. Space and time are split, which leads to new time evolution schemes for scalar wave equations. Issues like local time stepping and mesh refinement are discussed. Based on a variational formulation of the Einstein's equations, space-time schemes of finite differences, finite element, and discontinuous Galerkin schemes, are developed. The methods are evaluated for linear and non-linear test problems.