An efficient dissipation–preserving Legendre–Galerkin spectral method for the Higgs boson equation in the de Sitter spacetime universe

Abstract

Since the emergence of numerical methods in mathematical modeling, fundamental properties such as convergence, efficiency, computational accuracy, and stability have been considered crucial for deciding the utility of a numerical approach. In more recent years, various aspects of structure preservation have emerged as an important addition to these fundamental properties. For the Higgs boson equation in the de Sitter spacetime, the corresponding energy dissipation law has not been well studied on the discrete level. In this work, we address this issue. More precisely, we propose a finite difference/Galerkin spectral scheme that can inherit the discrete energy dissipation property. The fully discretized scheme combines the Galerkin-Legendre discretization in space and the Crank–Nicolson scheme in time. The fully discrete scheme preserves dissipation of energy when the spacetime is expanding, anti-dissipative of energy when the spacetime is contracting and conservation of energy when the spacetime is flat. The practical and fully detailed implementation of the method is presented. The stability, uniqueness and the convergence analyses are discussed, which show that the numerical scheme is unconditionally stable and convergent of second-order accuracy in time and optimal error estimate in space. Finally, some numerical simulations are performed to illustrate performance and versatility of the proposed method.