Bermúdez Sarmiento, Jason Josué

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    Método conservativo de diferencias finitas de alto orden para sistemas de ecuaciones de Gross-Pitaevskii en dos dimensiones
    (2023-05-10) Bermúdez Sarmiento, Jason Josué; Castillo, Paul; College of Arts and Sciences - Sciences; Santiago, Freddie; Ramos, Rafael; Department of Mathematics; Pérez Muñoz, Fernando
    A method based on high-order finite differences, motivated by the general idea of Lie-Trotter division methods, is presented to approximate the solution of a coupled system of N nonlinear Schrödinger/Gross-Pitaevskii equations in 2D. The solution obtained from this scheme exactly preserves the Hamiltonian of the system and a numerical invariant that can be interpreted as a perturbation of the total mass with respect to the time step. Under certain conditions, for example, when the Josephson junction is neglected or when the initial conditions satisfy certain algebraic properties, conservation of mass of the system is also achieved. Using a suitable class of high order symmetric finite difference approximations of the 2D Laplacian operator, it is shown that, for potentials with positive Hessian matrix, the directional division scheme of τ + h^{2p} order in the l_{2,h}-norm , for p = 1, 2, 3, 4. To achieve a high order in time, the proposed basic time step is combined with well-known composition methods, resulting in conservative methods of order τ^{2q} + h^{2p} with q = 1, 2, 3, 4. Conservation and precision are theoretically and numerically validated for model problems with and without internal Josephson atomic bonding, for problems in which the analytical solution is known and others in which a reference solution will be used, obtained from a sufficiently fine mesh.