zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Methods for the numerical solution of the nonlinear Schrödinger equation. (English) Zbl 0555.65061
The paper considers approximate solutions to the nonlinear Schrödinger equation $iu\sb t+\Delta u+\lambda \vert u\vert\sp{p-1}u=0$ together with two invariants for the solution u. Two methods are considered. The first is a finite difference scheme which the author calls a ”leap frog” scheme. For this method a bound for the difference between the approximate and exact solution is proved assuming that $k<C\sb 1$ and $h<C\sb 2$ where h and k are the grid lengths and $C\sb 1$ and $C\sb 2$ are positive constants. The second method may be described as a Crank-Nicolson-Galerkin method and has the additional advantage that the invariants are conserved by the approximate solutions. In this method differences are used for the time integration and finite elements are used for the space discretion. A parameter $h\in (0,1)$ is used as an index for the finite element functions and given that $k=o(h)$ where k is the step size in time a bound is derived for the error difference whenever $h<C\sb 1$ where $C\sb 1$ is a positive constant.
Reviewer: B.Burrows

65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
35Q99PDE of mathematical physics and other areas
81Q05Closed and approximate solutions to quantum-mechanical equations
Full Text: DOI