𝐴-stability and dominating pairs

Abstract

It is considered whether linear combinations of A-acceptable exponential approximations preserve the ^-stability, when the coefficients of the linear combination are selected in order to achieve exponential fitting. Various pairs of exponential approximations are discussed and the satisfactory pairs are characterized.

Local Linearization—Runge–Kutta methods: A class of A-stable explicit integrators for dynamical systems

“…This have become feasible due to advances in the computation of matrix exponentials (see, e.g., [26], [56], [17], [12], [25]) and multiple integrals involving matrix exponentials (see, e.g., [9], [59]). Some instances of this type of integrators are the methods known as exponential fitting [47], [10], [60], [8], [33], exponential integrating factor [46], exponential integrators [27], [31], exponential time differencing [13], [45], truncated Magnus expansion [36], [5], truncated Fer expansion [61] (also named exponential of iterated commutators in [35]), exponential Runge-Kutta [28], [29], some schemes based on versions of the variation of constants formula (e.g., [50], [37], [34], [51], [19]), local linearization (see, e.g., [53], [54], [41], [42], [7]), and high order local linearization methods [14], [16], [39], [32].…”