Advection-dominated accretion flows with causal viscosity

We present the basic equations and sample solutions for the steady-state global transonic solutions of the advection-dominated accretion flows (ADAFs) with a causal viscosity prescription. The procedures for the stable numerical calculations and all explicit formula for obtaining the solutions of ADAFs are also presented. We solve the transonic solutions of ADAFs by using the explicit numerical integrations, as in the Runge-Kutta method. In this calculation method, we first solve the physical values at the sonic radius where L'Hopital's rule is used. Then, we numerically solve the coupled differential equations of the radial velocity, the angular momentum, and the sound speed from the sonic radius in the inward and outward directions. By the calculation procedures presented in this paper, we can cover all parameter spaces of the transonic solutions of ADAFs. Sample transonic solutions for ADAF-thick disk and ADAF-thin disk are presented. The explicit formula for the analytical expansion around the singular points, the sonic and the viscous points, are presented. If we set the diffusion timescale to null, the formalism in this study becomes the formalism of the acausal viscosity that is usually used in past studies for calculating of the ADAF structure.