Quadrature algorithms to the luminosity distance with a time-dependent dark energy model

In our previous work [1], we have proposed two methods for computing the luminosity distance dLΛ in ΛCDM model. In this paper, two effective quadrature algorithms, known as Romberg Integration and composite Gaussian Quadrature, are presented to calculate the luminosity distance dLCPL in the Chevallier-Polarski-Linder parametrization(CPL) model. By comparing both the efficiency and accuracy of the two algorithms, we find that the second is more promising. Moreover, we develop another strategy adapted for approximating dLΛ in flat ΛCDM universe. To some extent, our methods can make contributions to the recent numerical stimulation for the investigation of dark energy cosmology.