Spectral Methods for Time‐dependent Studies of Accretion Flows. I. Two‐dimensional, Viscous, Hydrodynamic Disks

Abstract

We present a numerical method for studying the normal modes of accretion flows around black holes. In this first paper, we focus on two-dimensional, viscous, hydrodynamic disks, for which the linear modes have been calculated analytically in previous investigations. We use pseudospectral methods and low-storage Runge-Kutta methods to solve the continuity equation, the Navier-Stokes equation, and the energy equation. We devise a number of test problems to verify the implementation. These tests demonstrate the ability of spectral methods to handle accurately advection problems and to reproduce correctly the stability criteria for differentially rotating hydrodynamic flows. They also show that our implementation is able to handle sound waves correctly with nonreflective boundary conditions, to recover the standard solution for a viscous-spreading ring, and to produce correctly the Shakura-Sunyaev steady-disk solution. Finally, we have applied our algorithm to the problem of a nonaxisymmetric viscous-spreading ring and verify that such configuration is unstable to nonaxisymmetric perturbations.