  The study of many astrophysical flows requires computational algorithms that
can capture high Mach number flows, while resolving a large dynamic range in
spatial and density scales. In this paper we present a novel method, RAM: Rapid
Advection Algorithm on Arbitrary Meshes. RAM is a time-explicit method to solve
the advection equation in problems with large bulk velocity on arbitrary
computational grids. In comparison with standard up-wind algorithms, RAM
enables advection with larger time steps and lower truncation errors. Our
method is based on the operator splitting technique and conservative
interpolation. Depending on the bulk velocity and resolution, RAM can decrease
the numerical cost of hydrodynamics by more than one order of magnitude. To
quantify the truncation errors and speed-up with RAM, we perform one and
two-dimensional hydrodynamics tests. We find that the order of our method is
given by the order of the conservative interpolation and that the effective
speed up is in agreement with the relative increment in time step. RAM will be
especially useful for numerical studies of disk-satellite interaction,
characterized by high bulk orbital velocities, and non-trivial geometries. Our
method dramatically lowers the computational cost of simulations that
simultaneously resolve the global disk and well inside the Hill radius of the
secondary companion.

Instrumentation and Methods for Astrophysics;Earth and Planetary Astrophysics;Astrophysics of Galaxies;Solar and Stellar Astrophysics;Computational Physics