Energy conserving local discontinuous Galerkin methods for the nonlinear Schrödinger equation with wave operator (Q898499)

Considering the Schrödinger equation in a bounded, at most three-dimensional domain, the authors propose its numerical solution by taking the discontinuous Galerkin method in space and Crank-Nicolson in time. The boundary conditions are of first kind or periodic. Based, among others, on work by \textit{B. Dong} and \textit{C.-W. Shu} [SIAM J. Numer. Anal. 47, No. 5, 3240--3268 (2009; Zbl 1204.65123)], they prove that a basic property of their equation is also true for the semi-discretized (in space) equation: the conservation of energy. For the linear case of the equation and specializing the domain to a rectangular one, they further prove optimal error estimations for the semi-discretized equation considered on a finite time interval. Finally, taking Crank-Nicolson to fully discretize the equation, they show conservation of the discrete energy. In their numerical examples, illustrating the theoretical results, one- to three-dimensional equations are solved. But it remains unclear what are \(p^2\) polynomials and, on some of the figures, which part of the (complex) solutions is shown. The misprints to write Cauchy-Schwartz instead of Cauchy-Schwarz and Crank-Nicholson instead of Crank-Nicolson seem ineradicable.