An improved parallel meshless algorithm for two typical 2D/3D nonlinear dynamics equations

Abstract

In this work, an improved parallel pure meshless algorithm coupling with a high-order time split-step (HSS-IPFPM) is firstly developed to solve the multi-dimensional GPEs, and then the IPFPM is extended to investigate the multi-components CH equations on irregular domain. The proposed meshless method is mainly derived with three points: (a) a fourth-order time-splitting technique is adopted to decompose GPEs, and the high-order derivative is divided into multi-order derivatives; (b) the spatial derivatives are approximated by the Finite Pointset method (FPM) based on the Taylor expansion and weighted least square; (c) the MPI (Message Passing Interface) parallel technique is adopted to reduce the computational cost. Subsequently, the numerical stability and error estimation of the present method are discussed, and the convergent rate and calculated computation of the parallel algorithm are demonstrated by solving four examples. Then the present method is extended to predict inelastic collisions or quantum vortices in GPEs, and phase separation process in multi-component CH. Meanwhile, the merit of easy to implement non-uniformly distributed in the proposed meshless method is illustrated. Finally, the 2D/3D phase separation phenomena in irregular domain are numerically investigated to further demonstrate the capability and applicability of the proposed parallel meshless algorithm.

Keywords

• Gross-Pitaevskii equations
• High-order split-step
• Meshless method
• Parallelization
• Phase separation in Cahn-Hilliard