TT-M Finite Element Algorithm for the Coupled Schrödinger–Boussinesq Equations

Abstract

In this article, the coupled Schrödinger–Boussinesq equations are solved numerically using the finite element method combined with the time two-mesh (TT-M) fast algorithm. The spatial direction is discretized by the standard Galerkin finite element method, the temporal direction is approximated by the TT-M Crank–Nicolson scheme, and then the numerical scheme of TT-M finite element (FE) system is formulated. The method includes three main steps: for the first step, the nonlinear system is solved on the coarse time mesh; for the second step, by an interpolation formula, the numerical solutions at the fine time mesh point are computed based on the numerical solutions on the coarse mesh system; for the last step, the linearized temporal fine mesh system is constructed based on Taylor’s formula for two variables, and then the TT-M FE solutions can be obtained. Furthermore, theory analyses on the TT-M system including the stability and error estimations are conducted. Finally, a large number of numerical examples are provided to verify the accuracy of the algorithm, the correctness of theoretical results, and the computational efficiency with a comparison to the numerical results calculated by using the standard FE method.

A Conservative Difference Scheme for Solving the Coupled Fractional Schrödinger–Boussinesq System

“…Several methods for finding the exact solutions to the CSBS have been presented [7][8][9][10]. Because the exact solutions to the CSBS often contain certain special functions, scholars have begun utilizing efficient methods to seek numerical solutions for it, such as the multi-symplectic method [11], orthogonal spline collocation method [12], radial basis function-finite difference method [13], cut-off function method [14], scalar auxiliary variable method [15], Adams prediction-correction method [16], finite-element method [17,18], and energy-preserving compact finite difference methods [19].…”

A Time Two-Mesh Finite Difference Numerical Scheme for the Symmetric Regularized Long Wave Equation

“…A nonlinear distributed order diffusion model was efficiently solved using the TT-M algorithm in conjunction with the H 1 -Galerkin mixed finite element method by Wen et al [28], both smooth and non-smooth solutions were considered. Additionally, Tian et al [29] developed a finite element method equipped with the TT-M technique to solve the coupled Schrödinger-Boussinesq equations. Moreover, some studies have investigated combining the TT-M and finite difference methods to solve nonlinear fractional partial differential equations, such as the works of Qiu and Xu et al [30,31], who proposed and analyzed a TT-M algorithm based on finite difference methods.…”