An implicit and convergent method for radially symmetric solutions of Higgs' boson equation in the de Sitter space-time

摘要

The present work introduces a numerical scheme that preserves the dissipation of energy of the Higgs boson equation in the de Sitter space-time. More precisely, the model considered in this work is a mathematical generalization of Higgs' model which includes a general time-dependent diffusion coefficient and a generalized potential. The mathematical system is dissipative, and we propose an implicit discrete method which approximates consistently the radially symmetric solutions of the continuous system. At the same time, a discrete energy functional is presented, and we prove that, as its continuous counterpart, the numerical technique dissipates the energy of the discrete system. The properties of consistency, stability and convergence of the numerical model are proved rigorously. To confirm the theoretical results, we approximate some radially symmetric solutions of the classical Higgs boson equation in the de Sitter space-time. In particular, the numerical results confirm the stability and the formation of bubble-like solutions.