Split-step quintic uniform algebraic trigonometric tension b-spline collocation method for cubic Ginzburg-Landau equations

Abstract

This paper proposes a novel numerical framework for solving the one- and multi-dimensional cubic Ginzburg-Landau (CGL) equation by integrating the quintic Uniform Algebraic Trigonometric (UAT) tension B-spline collocation method with the Strang splitting technique. The approach decomposes the original equation into two nonlinear subproblems and one or more linear subproblems via a time-splitting strategy, achieving second-order temporal accuracy. The linear subproblems are resolved using the quintic UAT tension B-spline collocation method to ensure fourth-order spatial accuracy, while the nonlinear subproblems are solved analytically, forming an unconditionally stable scheme. The framework is extendable to other nonlinear PDEs, such as the Schrödinger equation, Kuramoto-Tsuzuki equation, and reaction-diffusion systems, enabling efficient simulations of complex systems in physics, engineering, and materials science. Numerical experiments and comparative analysis validate its accuracy, high convergence orders, and computational efficiency, establishing it as a new high-performance tool for solving such nonlinear PDEs.