MLegS: A modern mapped Legendre spectral method solver for unbounded domains with parallelization

Abstract

MLegS is an open-source Modern Fortran code package for solving linear and nonlinear time-dependent partial differential equations on radially unbounded domains. It builds on an earlier in-house prototype, recasting it as a modular solver with a unified interface for fields, transforms, and spectral operators. Within this framework, the numerical formulation maps $0\leq r<\infty$ onto a finite interval, where associated Legendre functions provide the radial basis, while Fourier series represent the periodic azimuthal and axial directions. The mapped radial basis represents radial infinity without imposing an artificial boundary condition at finite radius while incorporating the regularity required at $r=0$ according to azimuthal order. The implementation distributes the corresponding field layouts using the Message-Passing Interface (MPI), with strong- and weak-scaling measurements extending to 512 cores across four compute nodes. Beyond parallel performance, three examples assess the solver across diffusion, reaction-diffusion, and vortex dynamics: closed-form diffusion solutions establish spectral spatial convergence and the design orders of both semi-implicit integrators; a Fisher-Kolmogorov-Petrovsky-Piskunov front matches the linearized solution and propagates at the classical speed within its logarithmic correction; and vortex-pair simulations based on toroidal-poloidal decomposition yield a measured Crow growth rate $2.7\%$ above the theoretical prediction and demonstrate the reconnection of the vortex pair.