An efficient spectral Poisson solver for the nirvana-III code: the shearing-box case with vertical vacuum boundary conditions

TL;DR

This paper develops two spectral Poisson solvers for the Cartesian shearing-box with vertical vacuum boundary conditions to handle self-gravity in vertically stratified discs. SASHA combines an analytical mean-field term with a spectral solve for fluctuations, while VGF-HybridBC uses a regularized Green's function in Fourier space to perform a single-step 3D convolution; both leverage multi-dimensional FFTs and a coordinate mapping to a fully periodic frame. The methods achieve high accuracy with modest grid counts (third-order convergence for VGF-HybridBC and near-second to third-order for SASHA) and scale efficiently on large MPI runs, with spectral solver time under 6% of total runtime up to 4096 cores. Implemented in the Nirvana-III code and using pencil-decomposed FFTs (p3dfft), the work enables high-resolution gravito-turbulence and gravitational fragmentation studies in protoplanetary discs by providing fast, accurate, and boundary-condition-consistent Poisson solutions.

Abstract

The stability of a differentially rotating fluid subject to its own gravity is a problem with applications across wide areas of astrophysics--from protoplanetary discs (PPDs) to entire galaxies. The shearing box formalism offers a conceptually simple framework for studying differential rotation in the local approximation. Aimed at self-gravitating, and importantly, vertically stratified PPDs, we develop two novel methods for solving Poisson's equation in the framework of the shearing box with vertical vacuum boundary conditions (BCs). Both approaches naturally make use of multi-dimensional fast Fourier transforms for computational efficiency. While the first one exploits the linearity properties of the Poisson equation, the second, which is slightly more accurate, consists of finding the adequate discrete Green's function (in Fourier space) adapted to the problem at hand. To this end, we have revisited the method proposed by Vico et al. (2016) and have derived an analytical Green's function satisfying the shear-periodic BCs in the plane as well as vacuum BCs, vertically. Our spectral method demonstrates excellent accuracy, even with a modest number of grid points, and exhibits third-order convergence. It has been implemented in the NIRVANA-III code, where it exhibits good scalability up to 4096 CPU cores, consuming less than 6% of the total runtime. This was achieved through the use of P3DFFT, a fast Fourier Transform library that employs pencil decomposition, overcoming the scalability limitations inherent in libraries using slab decomposition. We have introduced two novel spectral Poisson solvers that guarantees high accuracy, performance, and intrinsically support vertical vacuum boundary conditions in the shearing-box framework. Our solvers enable high-resolution local studies involving self-gravity, such as MHD simulations of gravito-turbulence or gravitational fragmentation.

Figures (6)

• Figure 1: Potential associated with a one-dimensional vertical Gaussian distribution. We considered two cases: i) when centred around the midplane, and ii) with a shifted Gaussian profile.
• Figure 2: Potential along the z direction associated with a 3D density distribution: periodic in x and y, Gaussian in the vertical direction.
• Figure 3: Log-scale relative error cuts for the 3D test, employing the VGF-HybridBC method, with mixed boundary conditions. Top: $z=0$ plane. Bottom: $y=3$ plane.
• Figure 4: Convergence test: 1D (top) and 3D (bottom) results. The SASHA technique achieves second-order convergence, while the VGF-HybridBC method reaches slightly better than third-order convergence.
• Figure 5: Left: Characteristic times of oscillatory and collapse modes with respect to the wavelength. Right: Relative error.