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

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.