A strong-form local meshless approach based on radial basis function-finite difference (RBF-FD) method for solving multi-dimensional coupled damped Schrödinger system appearing in Bose–Einstein condensates

Abstract

In this work, one-dimensional (1D), two-dimensional (2D) and three-dimensional (3D) coupled damped Schrödinger system is solved numerically. A strong-form local meshless approach established on radial basis function-finite difference (RBF-FD) method for spatial approximation is developed. Polyharmonic splines are used as radial basis function with augmented polynomials. The use of the polyharmonic splines saves us from choosing an optimum shape parameter which is not a simple task for infinitely smooth RBFs such as multiquadrics or Gaussians. For time discretization classical fourth-order Runge Kutta method is utilized. L∞ error norm and conserved quantities are computed to indicate performance of the proposed method. Stability of the proposed method is examined numerically. Some computer codes are devised in Julia programming language for obtaining numerical results. Acquired numerical results and their comparison with other studies available in literature such as cubic B-spline Galerkin method and direct meshless local Petrov–Galerkin (DMLPG) method endorse the performance and reliability of the proposed method.