Documentation/UserGuide/Orbital Advection in cylindrical

The implementation of cylindrical coordinates in Athena includes an orbital advection, or FARGO, algorithm which greatly increases the time step in problems consisting of a supersonic background flow, for example geometrically thin accretion disks. Use of orbital advection can yield a speed-up of up to the maximum Keplerian Mach number of the simulation; in practice generally an order of magnitude for thin disks. Orbital advection is turned on by using the configure flag --enable-fargo. Currently, orbital advection is limited to the CTU integrator. The algorithm supports both second- and third-order reconstruction.

Properly setting up orbital advection in a cylindrical simulation is more difficult than the implementation for shearing box (Cartesian) simulations. First of all, two user-specified functions of the cylindrical radius are required, one representing the Keplerian angular velocity and the other representing the shearing parameter. The shear parameter, $q$, is given as $q = -0.5 * dln(\Omega^2)/dlnR$, where $\Omega = \Omega(R)$ is the Keplerian angular velocity profile. For example, given an unstratified Newtonian potential $\Phi = -1/R$, $\Omega(R) = R^{-1.5}$ and $q(R) = 1.5$. These functions would be defined similarly to a static gravitational potential, for instance:

static Real Omega(const Real R) {
  Real Arg;
  Arg = pow(R,1.5);
  return (1.0/Arg);
static Real Shear(const Real R) {
  return 1.5;

These functions need to be enrolled in the main body of the problem file as:

OrbitalProfile = Omega;
ShearProfile = Shear;

Once the function pointers OrbitalProfile and ShearProfile have been defined, the only remaining change from the standard cylindrical geometry is in the definition of the x2-momentum. When using orbital advection the x2-momentum is not meant to represent the full azimuthal velocity, but only the velocity in the locally-rotating frame. For a precisely Keplerian disk this means that the x2-momentum is identically zero. An example of proper initialization is given below.

#ifdef FARGO
       pG->U[k][j][i].M2 = 0.0;
       pG->U[k][j][i].M2 = pG->U[k][j][i].d*avg1d(vphi,pG,i,j,k);

A useful example of orbital advection is the problem file cylnewtmri.c in the standard Athena distribution. Worth noting, is that because the x2-momentum is stored as the value in the rotating frame it will also be output that way. This can sometimes require that the Keplerian velocity to be added back in using postprocessing scripts.