zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Linear instability implies spurious periodic solutions. (English) Zbl 0685.65088

A question of central importance in the numerical approximation of time evolving equations (reaction-diffusion-convection equations) is whether or not the simulation produces the same asymptotic behaviour as the underlying continuous problem, for fixed but small values of mesh- spacings. This question is closely related to the existence of spurious steady, periodic and quasi-periodic solutions generated by discretizations: linear instability (where the linearization is about any steady solution of the difference equations) implies the existence of spurious periodic solutions in the fully nonlinear problem.

A profound and fairly comprehensive analysis gives an answer to the question whether or not the periodic solutions can exist for arbitrarily small values of the time-step and what form they take. The method is a combination of local bifurcation theory and the study of modified singularly perturbed boundary value problems by singular perturbation techniques. Especially, a local construction, via bifurcation theory, of the spurious periodic solutions near to the critical values of time steps at which they bifurcate from the steady solution, is found. - Again, a thorough, exhausted study.

Reviewer: E.Lanckau

MSC:
65N12Stability and convergence of numerical methods (BVP of PDE)
35B25Singular perturbations (PDE)
35B32Bifurcation (PDE)
35K57Reaction-diffusion equations