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)
Time discretization of an integro-differential equation of parabolic type. (English) Zbl 0608.65096

The evolution equation u t +Au= 0 t κ(t,s)Bu(s)ds+f(t), t0, u(0)=v is considered for an unbounded positive-definite self-adjoint operator with dense domain D(A) in a Hilbert space and D(B)D(A). The kernel κ is assumed to be a smooth real-valued function. Stability and convergence are investigated for the backward Euler scheme and the Crank-Nicolson scheme, both for the integral supplemented by quadrature rules consistent with the mentioned difference schemes. For quadrature the interval [0,t n ] or [0,t n-1/2 ], respectively, is split into two subintervals: if k is the steplength of the difference scheme, the initial interval is discretized with a steplength k 1 =O(k 1/2 ), and the final interval taken to have a length <k 1 is discretized with steplength k. Then in the initial interval the trapezoidal rule or Simpson’s rule, respectively, is taken, in the final interval, however, simply the rectangle rule.

If everything is smooth enough the consistency order of the difference scheme is preserved, the advantage being sparseness of the quadrature nodal points. Modifications are possible in cases of not so good regularity properties. In applications to parabolic problems in space and time (A then being an elliptic differential operator) space discretization should be done in such a way that consistency order is not worsened. It is described how this can be achieved in the context of finite elements.

Reviewer: R.Gorenflo

MSC:
65R20Integral equations (numerical methods)
45K05Integro-partial differential equations