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On the stability of variable stepsize rational approximations of holomorphic semigroups. (English) Zbl 0801.65050

The paper deals with the stability of variable stepsize rational time approximations to the initial value problem \(u'(t) = Au(t)\); \(u(0) = u_ 0\), where \(A\) is the infinitesimal generator of a holomorphic semigroup. A rational method replaces the operator \(e^{tA}\) by an operator \(r({t \over k} A)^ k\), \(k \in \mathbb{N}\), with a rational approximation \(r(z)\) to the exponential. The Lax stability requires the boundedness of the powers \(r({t \over k} A)^ k\) for each \(t>0\). For strong \(A(\theta)\)- acceptability estimations are proved. The corresponding theorem could be useful in the theory of discretizations of ordinary differential equations when deriving stability bounds which are independent of the stiffness of the problem.

MSC:

65J10 Numerical solutions to equations with linear operators
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
34G10 Linear differential equations in abstract spaces
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