On the numerical experiments of the Cauchy problem for semi-linear Klein–Gordon equations in the de Sitter spacetime

Abstract

The computational analysis of the Cauchy problem for semi-linear Klein-Gordon equations in the de Sitter spacetime is considered. Several simulations are performed to show the time-global behaviors of the solutions of the equations in the spacetime based on the structure-preserving scheme. It is remarked that the sufficiently large Hubble constant yields the strong diffusion-effect which gives the long and stable simulations for the defocusing semi-linear terms. The reliability of the simulations is confirmed by the preservation of the numerically modified Hamiltonian of the equations.

Numerical simulations of semilinear Klein--Gordon equation in the de Sitter spacetime with structure-preserving scheme

“…For these results, the positiveness of the Hubble constant make stable and accurate simulations. We note that the same descriptions are in [1].…”

“…For numerical schemes of the partial differential equations, there are several well-known methods such as the Crank-Nicolson scheme and the Runge-Kutta scheme. Since the simulations with these schemes often have large numerical errors for the nonlinear equations (e.g., [1]), the suitable schemes have been suggested. One of the schemes is the structure preserving scheme (SPS) [2].…”

“…In this paper, we review the discretized equations of the semi-linear Klein-Gordon equation in the de Sitter spacetime with SPS and perform some simulations to confirm the stability and the numerical accuracy. For investigations of Klein-Gordon equation in the de Sitter spacetime, analytical researches [5,6,7,8,9] and numerical researches [10,1] are reported. In [1], we reported some stable and high-precision numerical results with SPS.…”

“…For investigations of Klein-Gordon equation in the de Sitter spacetime, analytical researches [5,6,7,8,9] and numerical researches [10,1] are reported. In [1], we reported some stable and high-precision numerical results with SPS. There are some differences between the standard discretized equation and the discretized equation with SPS.…”

The Cauchy problem for the Klein-Gordon equation under the quartic potential in the de Sitter spacetime

“…This result was further investigated on the semilinear term including the derivatives of the solution by Hintz and Vasy [16]. We refer to [26] on numerical computations for the semiliear Klein-Gordon equation, and [21,23] on the Cauchy problem for the semilinear Schrödinger equation and the semilinear Proca equation in the de Sitter spacetime.…”