Comparison of explicit and implicit difference schemes for parabolic functional differential equations

Zdzisław Kamont¹ ; Karolina Kropielnicka¹

¹ Institute of Mathematics University of Gdańsk Wit Stwosz Street 57 80-952 Gdańsk, Poland

Annales Polonici Mathematici, Tome 103 (2012) no. 2, pp. 135-160

Résumé

Initial-boundary value problems of Dirichlet type for parabolic functional differential equations are considered. Explicit difference schemes of Euler type and implicit difference methods are investigated. The following theoretical aspects of the methods are presented. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that the assumptions on the regularity of the given functions are the same for both methods. It is shown that the conditions on the mesh for explicit difference schemes are more restrictive than the suitable assumptions for implicit methods. There are implicit difference schemes which are convergent while the corresponding explicit difference methods are not convergent. Error estimates for both methods are constructed.

Détail

DOI : 10.4064/ap103-2-3

Keywords: initial boundary value problems dirichlet type parabolic functional differential equations considered explicit difference schemes euler type implicit difference methods investigated following theoretical aspects methods presented sufficient conditions convergence approximate solutions given comparisons methods presented proved assumptions regularity given functions methods shown conditions mesh explicit difference schemes restrictive suitable assumptions implicit methods there implicit difference schemes which convergent while corresponding explicit difference methods convergent error estimates methods constructed