We perform some simulations of the semilinear Klein--Gordon equation in the de Sitter spacetime. We reported the accurate numerical results of the equation with the structure-preserving scheme (SPS) in an earlier publication (Tsuchiya and Nakamura in J. Comput. Appl. Math. \textbf{361}: 396--412, 2019). To investigate the factors for the stability and accuracy of the numerical results with SPS, we perform some simulations with three discretized formulations. The first formulation is the discretized equations with SPS, the second one is with SPS that replaces the second-order difference as the standard second-order central difference, and the third one is with SPS that replaces the discretized nonlinear term as the standard discretized expression. As a result, the above two replacements in SPS are found to be effective for accurate simulations. On the other hand, the ingenuity of replacing the second-order difference in the first formulation is not effective for maintaining the stability of the simulations.
翻译:我们用空间时间对半线性克莱因-哥尔登半线性方程式进行一些模拟。我们在较早的出版物(Tsuchiya和Nakamura,载于J.Comput.Appl.Matth.\textbf{{361}:396-412,2019)中报告了与结构保护方案(SPS)等式的精确数字结果。为了用SPS调查数字结果的稳定性和准确性因素,我们用三种分解的配方进行一些模拟。第一个配方是用SPS分解的方程式,第二个配方是用SSPS取代二等差异,作为标准的二等中心差异,第三个配方是取代离散非线性非线性术语,作为标准的分解表达方式。因此,在SPS的上述两个替代方法被认为对精确模拟有效。另一方面,替代第一种配方的第二等差的巧妙性对于维持模拟的稳定性是无效的。