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

¹ Leibniz-Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam, Germany
² Niels Bohr International Academy, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen ∅, Denmark

^★ Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 11 October 2025
Accepted: 27 October 2025

Abstract

Context. 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 to entire galaxies. The shearing box formalism offers a conceptually simple framework for studying differential rotation in the local approximation.

Aims. Aimed at self-gravitating, and importantly, vertically stratified protoplanetary discs, we develop two novel methods for solving Poisson’s equation in the framework of the shearing box with vertical vacuum boundary conditions (BCs).

Methods. Both approaches naturally make use of multi-dimensional fast Fourier transforms (FFTs) 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 derived, in Fourier space, an analytical Green’s function satisfying the shear-periodic BCs in the plane as well as vacuum BCs, vertically.

Results. 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.

Conclusions. We have introduced two novel spectral Poisson solvers that guarantee high accuracy, performance, and intrinsically support vertical vacuum BCs in the shearing box framework. Our solvers enable high-resolution local studies involving self-gravity, such as magnetohydrodynamic (MHD) simulations of gravito-turbulence and/or gravitational fragmentation.