In this work, we investigate numerically Higgs’ boson equation in the de Sitter space-time, which is an important $(3+1)$-dimensional model arising in particle physics. Associated with this model, there exists a functional of energy which is dissipated with respect to time. Moreover, some relevant solutions of this model present radial symmetry in three spatial dimensions. In this work, we consider a general Hamiltonian form of this model and determine its expression in radial coordinates. The energy functional and the rate of change of the energy are also expressed in radial coordinates. Using a finite-difference approach, we propose a numerical method to approximate the solutions of the continuous model. At the same time, a discrete energy functional is proposed and we show that, as the total energy of the continuous model, the energy of the discrete system is dissipated with respect to time. We analyze the finite-difference scheme theoretically, and we prove that the discrete model and the discrete energy density are consistent discretizations of their continuous counterparts. We prove that the numerical model is stable and convergent, and provide some estimates of the numerical error. Some simulations on the formation of bubble-like solutions are provided for illustration purposes, and we provide comparisons against numerical schemes available in the literature.

在这项工作中，我们在de Sitter时空中对希格斯玻色子方程进行了数值研究，这很重要 $（3+\mathrm{1个}）$物理学中产生的三维模型。与该模型相关联，存在相对于时间消散的能量功能。此外，该模型的一些相关解决方案在三个空间维度上呈现径向对称性。在这项工作中，我们考虑该模型的一般汉密尔顿形式，并确定其在径向坐标中的表达。能量函数和能量变化率也以径向坐标表示。使用有限差分方法，我们提出了一种数值方法来近似连续模型的解。同时，提出了一个离散的能量函数，我们证明，作为连续模型的总能量，离散系统的能量相对于时间耗散。我们从理论上分析了有限差分方案，并且我们证明离散模型和离散能量密度是其连续对应项的一致离散化。我们证明了该数值模型是稳定且收敛的，并提供了一些数值误差的估计。为了说明起见，提供了一些关于气泡状溶液形成的模拟，并且我们与文献中提供的数值方案进行了比较。