Design and analysis of a dissipative scheme to solve a generalized multi-dimensional Higgs boson equation in the de Sitter space–time

Abstract

In this work, we design a numerically efficient finite-difference technique for the solution of a fractional extension of the Higgs boson equation in the de Sitter space–time. The model under investigation is a multidimensional equation with Riesz fractional derivatives of orders in (0,1)∪(1,2], which considers a generalized potential and a time-dependent diffusion factor. An energy integral for the mathematical model is readily available, and we propose an explicit and consistent numerical technique based on fractional-order centered differences with similar Hamiltonian properties as the continuous model. A fractional energy approach is used then to prove the properties of stability and convergence of the technique. For simulation purposes, we consider both the classical and the fractional Higgs real-valued scalar fields in the (3+1)-dimensional de Sitter space–time, and find results qualitatively similar to those available in the literature. The present work is the first paper to report on a Hamiltonian discretization of the Higgs boson equation (both fractional and non-fractional) in the de Sitter space–time and its numerical analysis. More precisely, the present manuscript is the first paper of the literature in which a dissipation-preserving scheme to solve the multi-dimensional (fractional) Higgs boson equation in the de Sitter space–time is proposed and thoroughly analyzed. Indeed, it is worth pointing out that previous efforts used techniques based on the Runge–Kutta method or discretizations that did not preserve the dissipation nor were rigorously analyzed.