TITLE

DIFFERENCE SCHEMES FOR NONLINEAR BVPS ON THE SEMIAXIS

AUTHOR(S)
Gavrilyuk, I. P.; Hermann, M.; Kutniv, M. V.; Makarov, V. L.
PUB. DATE
January 2007
SOURCE
Computational Methods in Applied Mathematics;2007, Vol. 7 Issue 1, p25
SOURCE TYPE
Academic Journal
DOC. TYPE
Article
ABSTRACT
The scalar boundary value problem (BVP) for a nonlinear second order differential equation on the semiaxis is considered. Under some natural assumptions it is shown that on an arbitrary finite grid there exists a unique three-point exact difference scheme (EDS), i.e., a difference scheme whose solution coincides with the projection of the exact solution of the given differential equation onto the underlying grid. A constructive method is proposed to derive from the EDS a so-called truncated difference scheme (n-TDS) of rank n, where n is a freely selectable natural number. The n-TDS has the order of accuracy n̄n = 2[(n + 1)/2], i.e., the global error is of the form O(|h|n̄), where |h| is the maximum step size and [·] denotes the entire part of the expression in brackets. The n-TDS is the basis for a new adaptive algorithm which has all the advantages known from the modern IVP-solvers. Numerical examples are given which illustrate the theorems presented in the paper and demonstrate the reliability of the new algorithm.
ACCESSION #
25068685

 

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