Métodos espectrais com decomposição de domínios: explorando o colapso gravitacional em espaços-tempos com simetria esférica e não-esférica

Abstract

Although published over a hundred years ago, the theory of General Relativity, which provides the classic description of gravitation, has few known exact solutions. For a more complete understanding of the implications of this theory, it is necessary that the solutions to the Einstein’s field equations to be pursued in another way. Numerical Relativity presents itself as an appropriate alternative for this purpose. In this work, the application of spectral methods to two problems of General Relativity is studied, emphasizing the use of the domain decomposition technique. The first problem addressed is the case of a spherically symmetric self-gravitating massless scalar field in the char acteristic formalism. The Galerkin-Collocation method is used in two, three and four subdomains. The codes are validated by convergence tests and as an application, Choptuik’s scale law of the critical phenomenon in gravitational collapse is reproduced. The second case studied is the Bondi problem with axial symmetry. Codes were developed in one and two domains, both based on the Galerkin-Collocation method and the Galerkin method with numerical integration. The convergence of the codes is verified and as a result the calculation of the Bondi mass for strong field regimes is presented. Bondi mass decay was also related to the extraction of gravitational waves through the news function. We sought to report in detail how strategies were adopted for the construction of the codes, their main advantages and disadvantages. It also stands out as possible applications that can be carried out and improvements to be performed.