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Fourth-order alternating direction implicit compact finite difference schemes for two-dimensional Schrödinger equations. (English) Zbl 1221.65220
The initial and boundary value problem in a bounded domain for two-dimensional linear and nonlinear Schrödinger equations with nonlinearity in a real-valued function at the linear term are numerically solved. By using the alternating direction implicit method fourth-order in space and second-order in time compact finite difference schemes on uniform meshes are constructed. At each time step the schemes are reduced to one-dimensional scale tridiagonal symmetric systems of algebraic equations, which is the well-known advantage of the considered numerical method. Theoretical results on stability and error estimates are obtained. In addition, six numerical examples are solved in details to illustrate the applicability of the presented finite difference schemes and the numerical results are compared with the exact solutions of the selected problems.
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35Q55NLS-like (nonlinear Schrödinger) equations
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