A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. (English) Zbl 1434.65207
Summary: We propose a Fourier pseudo-spectral scheme for the space-fractional nonlinear Schrödinger equation. The proposed scheme has the following features: it is linearly implicit, it preserves two invariants of the equation, its unique solvability is guaranteed without any restrictions on space and time step sizes. The scheme requires solving a complex symmetric linear system per time step. To solve the system efficiently, we also present a certain variable transformation and preconditioner.
MSC:
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
| 65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65F10 | Iterative numerical methods for linear systems |
| 35R11 | Fractional partial differential equations |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
| 81S40 | Path integrals in quantum mechanics |
Keywords:
fractional Schrödinger equation; structure-preservation; energy-preservation; pseudo-spectral method; bi-CGSTAB method; preconditioningReferences:
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