We propose explicit and conditionally stable combined numerical method based on using of both the conservative finite-difference scheme and non-conservative Rosenbrock method, for solving of 1D and 2D nonlinear Schrödinger equation. Each of these finite-difference schemes has own advantages and disadvantages. The conservative finite-difference scheme is implicit, conservative and possesses the property of asymptotic stability and the second order of approximation. The Rosenbrock method is conditionally conservative, explicit and possesses the same order of approximation in spatial coordinate only. Proposed finite-difference scheme is explicit and more effective for some cases.
The main idea of the combined method consists in using the Rosenbrock’s method near the boundaries of the domain. It means, that we introduce certain sub-domains near the boundaries of the domain under consideration. In other part of the domain, the conservative finite-difference scheme is used for computation. The problem solution is provided in several stages. In the first stage the problem solution is computed in the sub-domains at using the finite-difference scheme based on the Rosenbrock method with artificial boundary conditions (ABCs). Note, that for the first time layer we use the initial complex amplitude distribution. The second stage consists in the problem solution at using the conservative finite-difference scheme with boundary conditions (BCs) defined at the previous stage. Then, we repeat these stages by using the results obtained at using the conservative finite-difference scheme as the initial condition and using the results obtained on the base of Rosenbrock method as the BC for the solution computation on the next time layers.
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