Finite differences for coarse azimuthal discretization and for reduction of effective resolution near origin of cylindrical flow equations

In this paper, the errors generated by the computation of derivatives in the azimuthal direction θ when flow equations are solved in cylindrical coordinates using finite differences are investigated. They might be large for coarse discretizations even using high-order schemes, which led us to design explicit finite differences specially for 8, 16, 32 and 64 points per circle. These schemes are shown to improve accuracy with respect to standard finite differences, and to provide solutions for a two-dimensional propagation problem similar to those obtained using Fourier spectral methods in the direction θ. A method is then presented to alleviate the time-step limitation resulting from explicit time integration near cylindrical origin, when finite differences are used. It consists in calculating azimuthal derivatives at coarser resolutions than permitted by the grid, in the same way as usually done using spectral methods. In practice, a series of doublings of the effective discretization in θ is implemented. Thus simulations can for instance be performed on a grid containing nθ = 256 points with a time step 32 times larger, with an accuracy comparable to that achieved in corresponding simulations involving Fourier spectral methods in the direction θ.