A pseudo‐spectral method that uses an overlapping multidomain technique for the numerical solution of sine‐Gordon equation in one and two spatial dimensions

Abstract

In this article, we study an explicit scheme for the solution of sine-Gordon equation when the space discretization is carried out by an overlapping multidomain pseudo-spectral technique. By using differentiation matrices, the equation is reduced to a nonlinear system of ordinary differential equations in time that can be discretized with the explicit fourth-order Runge-Kutta method. To achieve approximation with high accuracy in large domains, the number of space grid points must be large enough. This yields very large and full matrices in the pseudo-spectral method that causes large memory requirements. The domain decomposition approach provides sparsity in the matrices obtained after the discretization, and this property reduces storage for large matrices and provides economical ways of performing matrix-vector multiplications. Therefore, we propose a multidomain pseudo-spectral method for the numerical simulation of the sineGordon equation in large domains. Test examples are given to demonstrate the accuracy and capability of the proposed method. Numerical experiments show that the multidomain scheme has an excellent long-time numerical behavior for the sine-Gordon equation in one and two dimensions.

A modified pseudospectral method for solving trajectory optimization problems with singular arc

“…So by using the mapping (33) and by noting that dt d D tkC1 tk 2 , the optimal control problem given in (23)- (27) is converted to the following minimization problem in the time domain OE 1, 1:…”

“…In recent years, pseudospectral methods have been extensively used for the numerical solution of engineering problems [25][26][27] and optimal control problems [24,28,29]. In the pseudospectral method, the state and control variables are approximated by interpolating polynomials with specific collocation points such as Legendre-Gauss-Lobatto, Legendre-Gauss, and LR points [24].…”

A numerical meshless method of soliton-like structures model via an optimal sampling density based kernel interpolation

“…One of the main advantages of meshless methods is that the selection of field nodes could be controlled automatically and adaptively in theory, but for most of existing methods of solving nonlinear PDE, the nodal distribution is preassigned [24,29,[38][39][40][41][42][43]. And it is clear that an adaptive selection of nodes will be more effective and accurate than a preassigned nodal distribution to capture the soliton structure.…”

Legendre spectral element method for solving sine-Gordon equation

“…Other numerical methods have also been used to solve the sine-Gordon equation such as Chebyshev tau meshless method [25], meshless method of lines [26], high-accuracy mul-tiquadric quasi-interpolation [27], reduced differential transform method [28], pseudospectral method [29], modified cubic B-spline differential quadrature method [30], modified cubic B-spline collocation method [31], etc.…”