On structure preserving and circulant preconditioners for the space fractional coupled nonlinear Schrödinger equations

Abstract

Summary When the implicit, conservative difference scheme with the fractional centered difference formula is employed to discretize the space fractional coupled nonlinear Schrödinger equations, in each time step, we need to solve a complex symmetric linear system. The real part of the coefficient matrix is a symmetric Toeplitz‐plus‐diagonal matrix, whereas the imaginary part is the identity matrix. In this paper, a structure preserving preconditioner and a circulant preconditioner are proposed for such Toeplitz‐like matrix. Theoretically, tight bounds for eigenvalues of the preconditioned matrices are derived. Numerical implementations show that Krylov subspace iteration methods such as BiCGSTAB, when accelerated by the proposed preconditioners, are efficient solvers for solving the discretized linear system.

Fourier spectral method with an adaptive time strategy for nonlinear fractional Schrödinger equation

“…Moreover, due to the nonlocal nature of the fractional operators, it is difficult to find out the analytic solutions for most FNLSs. Necessarily, many scholars devote to study the numerical solutions of FNLSs, and many effective numerical methods, such as finite difference method [19][20][21], finite element method [22,23], are employed to solve the FNLSs. Especially, the Fourier spectral methods [9,24,25] are more effective because of their high accuracy and stability in long-time behavior.…”

Editorial: Novel methods and theories in numerical algebra with interdisciplinary applications

“…Wang et al propose a structure‐preserving preconditioner and a circulant preconditioner for the Toeplitz‐like discretized linear systems arising from the space‐fractional coupled nonlinear Schrödinger equations, so that the Krylov subspace iteration methods such as BiCGSTAB can converge very quickly when used to solve the corresponding preconditioned linear systems.…”