Orbital Advection by Interpolation: A Fast and Accurate Numerical Scheme for Super-Fast MHD Flows

TL;DR

This paper introduces a fast, accurate numerical scheme for simulating super-fast magnetized astrophysical disks, reducing computational constraints and improving error consistency across the simulation domain.

Contribution

A FARGO-like algorithm for magnetized disks that is second order accurate, divergence-free, and suitable for efficient long-term simulations of super-fast MHD flows.

Findings

The scheme is second order accurate on smooth flows.

It preserves the divergence-free magnetic field to machine precision.

It reduces the variation of truncation error with position.

Abstract

In numerical models of thin astrophysical disks that use an Eulerian scheme, gas orbits supersonically through a fixed grid. As a result the time step is sharply limited by the Courant condition. Also, because the mean flow speed with respect to the grid varies with position, the truncation error varies systematically with position. For hydrodynamic (unmagnetized) disks an algorithm called FARGO has been developed that advects the gas along its mean orbit using a separate interpolation substep. This relaxes the constraint imposed by the Courant condition, which now depends only on the peculiar velocity of the gas, and results in a truncation error that is more nearly independent of position. This paper describes a FARGO-like algorithm suitable for evolving magnetized disks. Our method is second order accurate on a smooth flow and preserves the divergence-free constraint to machine precision. The main restriction is that the magnetic field must be discretized on a staggered mesh. We give a detailed description of an implementation of the code and demonstrate that it produces the expected results on linear and nonlinear problems. We also point out how the scheme might be generalized to make the integration of other supersonic/super-fast flows more efficient. Although our scheme reduces the variation of truncation error with position, it does not eliminate it. We show that the residual position dependence leads to characteristic radial variations in the density over long integrations.