A note on a conservative finite volume approach to address numerical stiffness in polar meshes

A polar coordinate system introduces a singularity at the pole, r=0, where terms with a factor 1/r can be ill-defined. While there are several approaches to eliminate this pole singularity in finite difference methods, finite volume methods largely bypass this issue by not storing or computing data at the pole. However, all methods face a very restrictive time step when using an explicit time advancement scheme in the azimuthal direction, where cell sizes are of the order O(Δr(rΔθ)). We use a conservative finite volume approach of merging cells on a structured O-mesh to remove this time step limit imposed by the CFL condition. The cell-merging procedure is implemented as a corrector step and incurs no changes to the underlying data structure for a structured grid. This short note describes the procedure and presents the validation and application of the algorithm to various problems. The algorithm is shown to be inexpensive and scalable. In addition, the cell-merging procedure is easily coupled with a line implicit scheme in the radial direction.