Abstract
This chapter is devoted to presenting the theory of linear stability for numerical methods solving ODEs. Characteristic properties occurring when the stepsize goes to zero do not reveal much of what happens for fixed values of the stepsize: linear stability theory is able provided insights regarding this issue. This theory relies on applying the numerical method to the so-called Dahlquist test problem and translates stability properties into stepsize restrictions. This analysis is given for multistep, multistage and multivalues numerical methods. Relevant notions (such as A-stability and L-stability) are given, also in connection with the role of Padé approximations in this theory. The analysis of the maximum attainable order of A-stable methods is given via the theory of order stars.
A method which cannot handle satisfactorily the linear test system is not a suitable candidate for incorporation into an automatic code. More precisely, linear stability theory provides a useful yardstick (if one can have a yardstick in the complex plane!) by which different linear multistep methods (or classes of such methods) can be compared as candidates for inclusion in an automatic code.
(John D. Lambert [ 242 ])
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D’Ambrosio, R. (2023). Linear Stability. In: Numerical Approximation of Ordinary Differential Problems . UNITEXT(), vol 148. Springer, Cham. https://doi.org/10.1007/978-3-031-31343-1_6
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