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W. Auzinger, R. Stolyarchuk:
"Rational multistep schemes for semilinear stiff systems";
Keynote Lecture: International Skorobogat'ko Mathematical Conference, Drohobych (invited); 2011-09-19 - 2011-09-23.

Let u'(t) = Au(t) + g(t,u(t)) represent a semilinear ODE stiff system emanating from the spatial discretization of an evolutionary PDE, where the linear part represents the dominant, stiff component. In computational physics one is interested in the efficient and accurate integration of such systems. If the linear part can be efficiently treated, for instance via spectral approximation in combination with an A-stable scheme, the question is how the full problem can be tackled in a similarly efficient way.

Since the 1970s, several authors have considered integration based on an A-stable rational approximations R(hA) to the matrix exponential exp{hA} in combination with a multistep ansatz for the nonlinear part. Later, exponential schemes have become popular where evaluation of $ e^{tA} $ is directly approximated using pseudospectral of Krylov subspace techniques.

We consider these approaches in a common framework. In particular, we demonstrate how rational schemes can be directly related to exponential schemes, which enables a precise convergence analysis for the rational versions, in particular for A-stable Padé approximations R(hA) to exp{hA}. We discuss implementation issues and present numerical examples.

Rational multistep schemes