On time-splitting spectral approximations for the Schrödinger equation in the semiclassical regime

Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

We consider time-dependent (linear and nonlinear) Schrodinger equations in a semiclassical scaling. These equations form a canonical class of (nonlinear) dispersive models whose solutions exhibit high-frequency oscillations. The design of efficient numerical methods which produce an accurate approxima- tion of the solutions, or at least of the associated physical observables, is a formidable mathematical challenge. In this article we shall review the basic analytical methods for dealing with such equations, including WKB asymp- totics, Wigner measure techniques and Gaussian beams. Moreover, we shall give an overview of the current state of the art of numerical methods (most of which are based on the described analytical techniques) for the Schrodinger equation in the semiclassical regime.

https://www.ki-net.umd.edu/pubs/files/FRG-2011.ActaNum.pdf

Mathematical and computational methods for semiclassical Schrödinger equations

We apply Wigner-transform techniques to the analysis of difference methods for Schrodinger-type equations in the case of a small Planck constant. In this way we are able to obtain sharp conditions on the spatial-temporal grid which guarantee convergence for average values of observables as the Planck constant tends to zero. The theory developed in this paper is not based on local and global error estimates and does not depend on whether caustics develop or not. Numerical test examples are presented to help interpret the theory.

Numerical approximation of quadratic observables of Schrödinger-type equations in the semi-classical limit

In the 20th century, an important contribution to the modeling of wave propagation prediction is the Parabolic Equation (PE) approximation method. This did not happen until the last quarter of the century. During that time period, a number of contributions were made in the first 15 years, then, contributions were made continuously until the end of the 20th century. Contributions to the PE method are still going on. During the second half of the 1970 when the PE was introduced to the field of underwater acoustics, its main purpose was to predict long-range, low-frequency acoustic propagations under range-dependent environments in fluid medium; thus, there were a number of limitations. As time progresses these limitations were somewhat relaxed a great deal gradually due to many useful contributions. Up to this date, all contributions are aimed at the enhancement of the capability of the PE method. A few review papers of the PE have been presented in open archived literatures. This paper gives a complete survey of what has been done from the start until the end of 1999. Contributions up to the year 1994 were already reported in some detail in archived literatures; those contributions will be outlined. Surveying the contributions from 1995 to 1999 is a new addition to the already published survey papers. In recent years, the PE method has been used for many real applications; some results are included in this paper. A discussion will be given at the end of this paper on looking ahead how much more the PE can do in order to stimulate future research, development, and applications.

Parabolic equation development in the twentieth century

Most conventional explicit finite difference schemes, e.g. Euler’s scheme, for solving the parabolic equation of Schrodinger type $u_t = iu_{xx} $ are unconditionally unstable. This difficulty can be overcome by introducing a dissipative term to the conventional explicit schemes. Based on this approach, we derive a class of new explicit finite difference schemes which are conditionally stable, spans two time levels and are $O(k,h^2 )$ accurate. We also determine the schemes from this class that have the least restrictive stability requirements. It is interesting to note that the analog of the Lax–Wendroff scheme is unstable.

Stable explicit schemes for equations of the Schro¨dinger type

A class of ocean acoustic wave propagation problems is represented by a parabolic equation of the Schrodinger type. Using conventional explicit finite difference schemes, e.g. the Euler scheme, to solve the parabolic wave equation is unstable. Thus, important advantages of explicit schemes are completely missing. This paper presents a conditionally stable explicit scheme by introducing an extra dissipative term. This new explicit scheme is then applied to solve the ocean acoustic parabolic wave equation fully utilizing the advantages of explicit schemes. The theoretical development, the computational aspects, and the advantages are discussed. Application of the scheme to a realistic ocean acoustic problem is included. The solution obtained is compared with the unconditionally stable Crank-Nicolson solution.

https://apps.dtic.mil/sti/pdfs/ADA141800.pdf

A Stable Explicit Scheme for the Ocean Acoustic Wave Equation.

We introduce and analyze a collection of difference schemes for the numerical solution of a model multidimensional equation of Schrodinger type with applications to the three-dimensional parabolic wave equation arising from the sound propagation in the ocean. This collection of methods includes explicit and implicit schemes, two-level and three-level schemes and real and complex schemes. Many of these are analogous to classical schemes for the heat equation and the wave equation, but some schemes are unique to the Schrodinger equation. Von Neumann type stability results are given for all schemes. Numerical results arising from the application to an ocean acoustic problem are presented.

Difference schemes for the parabolic wave equation in ocean acoustics

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Stable explicit schemes for equations of the Schro¨dinger type

A Stable Explicit Scheme for the Ocean Acoustic Wave Equation.

Difference schemes for the parabolic wave equation in ocean acoustics