Relaxing Numerical Difficulties Arising from Discretization in the Cylindrical Coordinates

Abstract

The cylindrical coordinates, (r, ϕ, z), are often used in our numerical simulations, in particular, when gas accretes onto a central object. Although the cylindrical coordinates have several advantages in describing rotation, they have apparent singularity at r = 0. The singularity induces some difficulties in numerical simulations. First, it is difficult to solve the flow across the axis numerically. Second, the time interval between two successive time steps is restricted by the innermost cell size, resulting in large numbers of time steps to follow dynamics in the outer regions. Here, we present a new discretization scheme to overcome these difficulties. First, we evaluate the centrifugal force from the dynamical pressure working on the azimuthal cell surface. Our method of the evaluation is based on the vector analysis. Second, we reduce the angular resolution appreciably. Our examples demonstrate that the innermost circular region around the axis can be resolved by only six numerical cells. We increase the angular resolution with the increase in the radius so that each numerical cell has an aspect ratio close to unity. Then all the numerical cells have nearly the same size, and the CFL condition is relaxed. We show our application to an accreting protostar after showing a Sod shock tube problem and a uniform flow. The test problems demonstrate that the apparent singularity around the axis have no critical effects on the results.