Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime

Yazici, Muhammet; Şengül, Süleyman

doi:10.1515/phys-2016-0037

Cited by 10 publications

(3 citation statements)

References 19 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Tsuchiya¹,

Nakamura²

2022

Preprint

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We perform some simulations of the Klein-Gordon equation in the de Sitter spacetime. We reported the stable and high-precision numerical results of the equation using the structure preserving scheme (SPS) in an earlier publication (Tsuchiya and Nakamura in J. Comput. Appl. Math. 361: 396-412, 2019). To investigate the factors of the stability and accuracy of the simulations with SPS, we perform some simulations with three discretized formulations. The first formulation is the discretized equations with SPS, the second is with SPS which replaced the second derivative term as the standard difference, and the third is with SPS which replaced the non-linear term as the standard expression. As a result, the above two replacements in SPS are found to be effective for stable and high-precision simulations.

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Section: Introductionmentioning

confidence: 99%

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

Tsuchiya¹,

Nakamura²

2022

Preprint

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“…It is reported that the simulations of the nonlinear Klein-Gordon equation in the flat spacetime with the method [5]. In [31], the differential transform method [32] and the variational iteration method [24] are used to perform the nonlinear Klein-Gordon equation in the de Sitter spacetime. The former is one of the power series expansions.…”

Section: Introductionmentioning

confidence: 99%

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

Tsuchiya

Nakamura

2019

Journal of Computational and Applied Mathematics

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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.

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“…Comparison of several explicit finite difference methods have been presented for the one spatial variable case in [15]. A differential transform method with variational iteration method with Adomian's polynomials was presented very recently for the Higgs boson equation in de Sitter Spacetime in [30]. The Adomian decomposition method was also used for the Klein-Gordon equation with quadratic nonlinearity in [3] with a general method presented for the three space dimensional case and an example with a known solution presented for a one space dimensional case.…”

Section: Introductionmentioning

confidence: 99%

Computational Results for the Higgs Boson Equation in the de Sitter Spacetime

Balogh¹,

Banda²,

Yagdjian³

2018

Preprint

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High performance computations are presented for the Higgs Boson Equation in the de Sitter Spacetime using explicit fourth order Runge-Kutta scheme on the temporal discretization and fourth order finite difference discretization in space. In addition to the fully three space dimensional equation its one space dimensional radial solutions are also examined. The numerical code for the three space dimensional equation has been programmed in CUDA Fortran and was performed on NVIDIA Tesla K40c GPU Accelerator. The radial form of the equation was simulated in MATLAB. The numerical results demonstrate the existing theoretical result that under certain conditions bubbles form in the scalar field. We also demonstrate the known blow-up phenomena for the solutions of the semilinear Klein-Gordon equation with imaginary mass. Our numerical studies suggest several previously not known properties of the solution for which theoretical proofs do not exist yet: 1. smooth solution exists for all time if the initial conditions are compactly supported and smooth; 2. under some conditions no bubbles form; 3. solutions converge to step functions related to unforced, damped Duffing equations.

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Approximate solutions to the nonlinear Klein-Gordon equation in de Sitter spacetime

Cited by 10 publications

References 19 publications

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

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

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

Computational Results for the Higgs Boson Equation in the de Sitter Spacetime

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