A second-order \(L2\)-\(1_\sigma\) difference scheme for the nonlinear time-space fractional Schrödinger equation. (English) Zbl 1533.65133
Summary: In this paper, we mainly consider the numerical approximation for the nonlinear time-space fractional Schrödinger equation with the Dirichlet boundary condition. A linearized second-order finite difference scheme is presented by adopting the \(L2\)-\(1_\sigma\) formula to approximate the Caputo fractional derivative and the second-order weighted and shifted Grünwald difference formula to approximate the Riesz fractional derivative. To our knowledge, obtaining numerical solutions in the numerical scheme with the nonlinear term is very expensive. In order to reduce the computational cost and memory, the nonlinear term is handled by the local extrapolation method, which is more efficient than implicit processing and without reduced the convergence accuracy. Meanwhile, the numerical scheme of the coupled nonlinear time-space fractional Schrödinger system is also presented. The optimal error estimate of the numerical solution with convergence order \(O(\tau^2+h^2)\) is strictly proven. Finally, a series of numerical experiments are presented to confirm the accuracy and effectiveness of the proposed numerical schemes.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65B05 | Extrapolation to the limit, deferred corrections |
| 26A33 | Fractional derivatives and integrals |
| 35R11 | Fractional partial differential equations |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
Keywords:
time-space fractional Schrödinger equation; Caputo fractional derivative; Riesz fractional derivative; \(L2\)-\(1_\sigma\) formula; optimal error estimateReferences:
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