Because PPDs are initially quite massive in comparison to their host star, their own weight can lead to gravitational instability (GI) and subsequent fragmentation. The interaction of GI with magnetic fields is still poorly understood. To study these topics, we extend the NIRVANA magnetohydrodynamics code to the realm of self-gravity — specifically for the case of spherical-polar meshes. These are mandatory to properly keep track of the disk angular moment content, which in turn critically determines protoplanetary disk evolution. An obstacle is posed by the fact that solving Poisson's equation requires prior knowledge of the gravitational potential on the domain boundary. Standard multipole-expansion is shown to fail for the inner cavity in the polar mesh. As a remedy, and similar to Baron Münchhausen, who pulls himself out of the swamp by his own hair, we use James' trick, employing the already existing multi-grid solver to obtain a surface correction density (i.e. for the preliminary situation where a vanishing gravitational potential is assumed on the boundary/periphery). Translating this "screening charge" into the surface gravitational potential is done via a Green's function convolution method. We describe a novel method to obtain the required discrete kernel function in spherical polar geometry. Care has to be taken to obtain proper convergence with numerical resolution. For this purpose, we devise a new test-problem and provide a semi-analytic solution. Moreover, we demonstrate decent scaling behaviour of the implementation, up to several thousand independent compute cores.
Moreover, PPDs are often found to be extremely thin with respect to their vertical extent. Thus, in a complementary approach, we aim to describe the effects of self gravity (SG) in a vertically-integrated two-dimensional description — enabling highly resolved simulations at moderate computational cost. To this end, our research team pioneers a novel kernel-based framework, that greatly supersedes previous attempts using a simple softening-parameter. We demonstrate that — depending on the ad-hoc choice of that parameter — that class of approximation is bound to either severely under-estimate the effect of SG on certain length scales, or conversely, over-estimates it by equal amounts. We have developed an efficient implementation of the kernel-based two-dimensional approach, and have carefully benchmarked its accuracy against a three-dimensional reference solution. Employing this new framework, we will investigate the clumping of gravitationally unstable gas under various cooling regimes, with the aim to firmly establish the conditions that demarcate the boundary towards fragmentation.
