A Fully Pseudospectral Scheme for Solving Singular Hyperbolic Equations on Conformally Compactified Space-Times

Abstract

Abstract. With the example of the spherically symmetric scalar wave equation on Minkowski space-time we demonstrate that a fully pseudospectral scheme (i.e. spectral with respect to both spatial and time directions) can be applied for solving hyperbolic equations. The calculations are carried out within the framework of conformally compactified space-times. In our formulation, the equation becomes singular at null infinity and yields regular boundary conditions there. In this manner it becomes possible to avoid "artificial" conditions at some numerical outer boundary at a finite distance. We obtain highly accurate numerical solutions possessing exponential spectral convergence, a feature known from solving elliptic PDEs with spectral methods. Our investigations are meant as a first step towards the goal of treating time evolution problems in General Relativity with spectral methods in space and time.

A spectral solver for evolution problems with spatial -topology

“…Such a discretization technique for various spatial domains has a long history in computational physics, see for instance the references and examples in [6]. An alternative approach, which uses spectral discretization both in space and in time, has been reported on in [23]. However, to our knowledge, the case of spatial S 3 -topology has not been studied yet.…”

Axisymmetric fully spectral code for hyperbolic equations

“…This assumptions is strongly confirmed through several numerical experiments. 5 According to [24], γ is a root of the polynomial…”

“…As described in [5,7,8], this feature is due to the implicitness of the scheme, realized in both the fully pseudo-spectral and the SDIRK-method. In particular, we were able to take relatively large time steps (∆τ ∼ 100) with a small number of grid points (n τ ∼ 10), and, at the same time, a high resolution in the radial direction (n σ ∼ 100 for an interval ∆σ ∼ 1).…”

“…Email addresses: rodrigo.panosso-macedo@uni-jena.de (Rodrigo Panosso Macedo), marcus.ansorg@uni-jena.de (Marcus Ansorg) In this work, we extend the results of [5,7,8] and introduce a fully spectral code for axisymmetric hyperbolic PDEs. In particular, we address two limitations of the code presented in [8].…”

“…to both space and time. To the best of our knowledge, a first study along this line was performed for parabolic PDEs [4], while a fully spectral code for hyperbolic equations was first proposed in [5]. These works were restricted to problems with one spatial and one time dimension.…”