Shadowing for nonautonomous parabolic problems with applications to long-time error bounds

Citation
A. Ostermann et C. Palencia, Shadowing for nonautonomous parabolic problems with applications to long-time error bounds, SIAM J NUM, 37(5), 2000, pp. 1399-1419
Citations number
14
Categorie Soggetti
Mathematics
Journal title
SIAM JOURNAL ON NUMERICAL ANALYSIS
ISSN journal
00361429 → ACNP
Volume
37
Issue
5
Year of publication
2000
Pages
1399 - 1419
Database
ISI
SICI code
0036-1429(20000526)37:5<1399:SFNPPW>2.0.ZU;2-H
Abstract
Shadowing results for time discretizations of nonautonomous parabolic evolu tion equations z'(t) = A(t)z(t) + f(t) are given. The analysis is performed in an abstract Banach space setting of uniformly sectorial operators A ( t ). Under natural assumptions on the operators A ( t), it is shown that the backward Euler discretization of a smooth solution, obtained with constant stepsize h, is Ch-shadowed by a true solution of the system and, vice versa , every smooth trajectory is Ch-shadowed by a suitable backward Euler discr etization. This validates numerical computations over long time intervals. It is further explained how these results extend to higher order Runge-Kutt a methods.