Long-term stability of variable stepsize approximations of semigroups

Abstract

Abstract. This paper is concerned with the stability of rational one-step approximations of C 0 semigroups. Particular emphasis is laid on long-term stability bounds. The analysis is based on a general Banach space framework and allows variable stepsize sequences. Under reasonable assumptions on the stepsize sequence, asymptotic stability bounds for general C 0 semigroups are derived. The bounds are typical in the sense that they contain, in general, a factor that grows with the number of steps. Under additional hypotheses on the approximation, more favorable stability bounds are obtained for the subclass of holomorphic semigroups.

A functional calculus approach for the rational approximation with nonuniform partitions

“…By Theorem 5.1, we know that α c ≤ 1/2. Besides, according to [BO,Corollary 2.1] P n (A) ≤ C ln(1 + τm τ * )n 1/2 . Thus using also (5.10), we obtain for u and v as in Lemma 5.5,…”

A-stable Runge–Kutta methods for semilinear evolution equations

“…In [21], quoted above, Hersh and Kato prove convergence of A-acceptable rational approximations S(hA) of non-analytic C 0 -semigroups e hA for smooth initial data. Brenner and Thomée [6] show that A-acceptable rational approximations S(hA) of non-analytic C 0 -semigroups e hA with Re(spec A) ≤ 0 in general grow like S n (hA) = O(n 1/2 ) and study fractional order convergence for non-smooth initial data of linear evolution equations, see also [22]; for extensions to variable step size, see [4]. Brenner et al [7] study convergence of rational approximations of inhomogeneous linear differential equations on Banach spaces, assuming stability of the approximation.…”

On the Crank-Nicolson scheme once again

“…Since j=1, τ j ≤1 τ j < ∞ we know by Theorem 4.3 that y := lim k→∞ A (1) k x exists, and by what is proved above we know that lim →∞ A (2) k y exists for some subsequence (A (2) k ) ∈N of the sequence (A (2) k ) k∈N . This shows that for every x ∈ X the sequence (A k x) k∈N has a convergent subsequence.…”

“…For more details on strongly A-stable rational functions r (z), i.e. rational functions r (z) for which |r (z)| ≤ 1 for z ≤ 0 and lim sup |z|→∞ |r (z)| < 1, the reader is referred to [1] and [15]. For a related paper, see [10].…”