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
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.