Geometric Integrators for the Nonlinear Schrödinger Equation

Author： Islas A.L. Karpeev D.A. Schober C.M.

ISSN： 0021-9991

Source： Journal of Computational Physics, Vol.173, Iss.1, 2001-10, pp. : 116-148

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Abstract

Recently an interesting new class of PDE integrators, multisymplectic schemes, has been introduced for solving systems possessing a certain multisymplectic structure. Some of the characteristic features of the method are its local nature (independent of boundary conditions) and an equal treatment of spatial and temporal variables. The nonlinear Schrödinger equation (NLS) has a multisymplectic formulation, and in this paper we discuss the performance of both symplectic and multisymplectic integrators for the NLS. In the numerical experiments, the multisymplectic concatenated midpoint scheme (a centered cell discretization) is shown to preserve the local conservation laws extremely well over long times and to preserve global invariants such as the norm and momentum within roundoff. On the other hand, an integrable Hamiltonian semi-discretization of NLS from Ablowitz and Ladik (AL) possesses a full set of global conservation laws and a noncanonical symplectic structure. We generalize the generating function technique to develop symplectic integrators of arbitrary order for a general class of noncanonical systems carrying a symplectic structure of the AL type. Another approach examined in the paper is the introduction of transformations to reduce the AL system to either (1) separable form or (2) canonical form and then apply standard schemes in the new coordinates. All of the discretizations are tested numerically using initial data for spatially periodic multiphase solutions. The performance of the schemes as well as interrelations among various geometric features are discussed.