Using moving meshes helps reduce the angular momentum errors from advection in grid codes. We have run >200 iterations of this test problem using the public version of fvmhd3d, systematically varying choices like the mesh regularization scheme, mesh ‘drifting’ (whether to use a strictly Lagrangian drift, or locally smoothed velocity, or regularized drift), initial mesh geometry, and boundary conditions. In both fvmhd3d and more limited tests with arepo, we find that running in the ‘simplest’ initial configuration (an initial Cartesian mesh with outflow boundary conditions, with the default mesh regularization scheme used for all other test problems shown here), the disc goes unstable and the angular momentum evolution tends to be corrupted within a few orbits (similar to the fixed-grid cases). Unfortunately, some significant errors in angular momentum evolution are difficult to avoid in moving-mesh codes, as has been discussed extensively in e.g. Duffell & MacFadyen (2012), Ivanova et al. (2013), Mocz et al. (2014) and Muñoz et al. (2014). In a shearing disc, if the cells adapt in a truly Lagrangian manner, then they are inevitably deformed into a highly sheared/irregular shape (Muñoz et al. 2014). This leads to other errors. As soon as they become non-spherical (or more accurately fail to be radially symmetric about their own cell centre of mass), then mass advection between cells necessarily leads to additional angular momentum errors (indeed, the angular momentum of an irregular cell cannot be defined exactly but only to the same order of integration accuracy as the local velocity gradient estimator). This is akin to the errors in our MFV method. More importantly, if some regularization procedure is used to keep the cell shapes ‘regular’ (as is necessary in any moving-mesh code used for this problem), then the regularization means the cells cannot move entirely with the fluid and the gas must be advected over the cells. This re-introduces some of the same (more serious) errors we saw with stationary-grid methods (specifically, see Ivanova et al. 2013, equation 53). This means that the results for moving meshes are quite sensitive to choices like the mesh ‘stiffness’, regularization procedure, and in particular the choice of boundary conditions for the mesh-generating points (since the rigid Voronoi volume partition can lead to a ‘mesh tension’ effect, whereby regularization-induced distortions in the central regions propagate outwards ‘through’ the mesh; Springel 2010). So there are ways to improve the situation on this problem – for this reason, we do not show a single ‘standard’ moving-mesh result, because significantly different results are obtained if we make just small changes to the mesh-regularization procedure in each code. However, like with AMR codes, the most effective methods for eliminating angular momentum errors in moving meshes generally depend on knowing the problem geometry ahead of time. For example, Duffell & MacFadyen (2012) design a moving grid which is a series of cylindrical shells free to rotate independently about a shared axis (the disco code); Muñoz et al. (2013) use a carefully chosen initial grid configuration with a specially designed boundary condition designed to prevent inward propagation of ‘mesh deformation’; these help considerably, but must be fine-tuned to the exact disc configuration.
Citation: Muñoz et al. 2014
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