Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions. (English) Zbl 1349.65347
Summary: In this paper, a fourth-order compact and energy conservative difference scheme is proposed for solving the two-dimensional nonlinear Schrödinger equation with periodic boundary condition and initial condition, and the optimal convergent rate, without any restriction on the grid ratio, at the order of \(O(h^4+\tau^2)\) in the discrete \(L^2\)-norm with time step \(\tau\)and mesh size \(h\) is obtained. Besides the standard techniques of the energy method, a new technique and some important lemmas are proposed to prove the high order convergence. In order to avoid the outer iteration in implementation, a linearized compact and energy conservative difference scheme is derived. Numerical examples are given to support the theoretical analysis.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
nonlinear Schrödinger equation; compact and conservative difference scheme; a priori estimate; unconditional convergenceReferences:
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