A medium-shifted splitting iteration method for a diagonal-plus-Toeplitz linear system from spatial fractional Schrödinger equations. (English) Zbl 1499.35693
Summary: The centered difference discretization of the spatial fractional coupled nonlinear Schrödinger equations obtains a discretized linear system whose coefficient matrix is the sum of a real diagonal matrix \(D\) and a complex symmetric Toeplitz matrix \(\tilde{T}\) which is just the symmetric real Toeplitz \(T\) plus an imaginary identity matrix \(iI\). In this study, we present a medium-shifted splitting iteration method to solve the discretized linear system, in which the fast algorithm can be utilized to solve the Toeplitz linear system. Theoretical analysis shows that the new iteration method is convergent. Moreover, the new splitting iteration method naturally leads to a preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are tighter than those of the original coefficient matrix \(A\). Finally, compared with the other algorithms by numerical experiments, the new method is more effective.
MSC:
| 35R11 | Fractional partial differential equations |
| 65F10 | Iterative numerical methods for linear systems |
| 26A33 | Fractional derivatives and integrals |
| 35A15 | Variational methods applied to PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
spatial fractional Schrödinger equations; Toeplitz matrix; medium-shifting iteration method; convergence; preconditioningReferences:
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