2 Notations and standard method

We consider a polar grid composed of $N_{\rm s}$ sectors, each one $\Delta\theta=\frac{2\pi}{N_{\rm s}}$ wide, and $N_{\rm r}$ rings, with separations at radii $R_{i_{(0\leq i\leq N_{\rm r})}}$. The inner boundary is then located at the radius R₀, and the outer one at the radius $R_{N_{\rm r}}$. The density (and the internal energy if needed by the equation of state) is centered in the cells, and is denoted $(\Sigma_{ij})_{(i,j)\in[0,N_{\rm r}-1]\times[0,N_{\rm s}-1]}$. The radial velocity is denoted $v^{\rm r}_{ij}$, and is considered centered in azimuth and half-centered in radius (applied at radius R_i, i.e. at the interface between the cells [i,j] and [i-1,j]). In a similar way, the azimuthal velocity is denoted $v^\theta_{ij}$, and is considered centered in radius and half-centered in azimuth (i.e. at the interface between the cells [i,j] and [i,j-1]; throughout this paper the algebra on the j coordinate is meant in $Z/N_{\rm s}Z$to account for the periodicity in azimuth). Usually in a finite difference code the timestep is split in two main parts (Stone & Norman 1992). The first part is composed of eulerian substeps which consist in updating the HD quantities through the source terms in the evolution equations, and which include all the physical processes at work: pressure, gravity, viscosity, etc., and which can formally be described by the transformation $\xi \stackrel{E}{\rightarrow}\xi^a$, $\xi$ being any HD field on the grid. The second part is the transport substep, in which the quantities are conservatively moved through the grid according to the flow $[(v_{ij}^{\rm r})^a, (v_{ij}^\theta)^a]$, and which can be formally represented as $\xi^a\stackrel{R}{\rightarrow}\xi^b\stackrel{T}{\rightarrow}\xi^+$, where $\xi^+$ denotes any HD field after a whole timestep is completed, and R and T denote respectively the radial and azimuthal transport operators, which can be alternated every other timestep. The CFL condition comes both from the source part and the transport part, and the most stringent restriction is given by the T-substep, due to the unperturbed azimuthal flow. Classically, the azimuthal transport can be written as:

$$\xi_{ij}^+=\xi_{ij}^b+\frac{\Delta t}{\Delta y_i} \left(\xi_... ...ta a}- \xi_{ij+1}^{b,*/v^{\theta a}}v_{ij+1}^{\theta a}\right)$$ (1)

where $\Delta y_i=\frac{R_i+R_{i+1}}{2}\Delta\theta$ is the "mean azimuthal width'' of a cell. Equation (1) expresses the balance of the arbitrary conservative quantity $\xi$ in the cell [i,j]by computing the difference of its inflow at the [i,j-1]/[i,j] interface with the velocity $v_{ij}^{\theta a}$ and its outflow at the [i,j+1]/[i,j] interface with the velocity $v_{ij+1}^{\theta a}$. Actually we consider the flux of the upwinded interfacial quantity $\xi^{b,*/v^{\theta a}}$, where the " $*/v^{\theta a}$'' operator depends on the numerical method (donor cell, van Leer, PPA, see e.g. Stone & Norman 1992) and on the velocity field $v^{\theta a}$.