Two-Parameter, Arbitrary Order, Exponential Approximations for Stiff Equations

Abstract

Abstract.A two-parameter family of approximations to the exponential function is considered. Constraints on the parameters are determined which guarantee the approximations are ^-acceptable. The suitability of these approximations for 2-point vl-stable exponential fitting is established. Several numerical methods, which produce these approximations when solving y = \v, are presented.

𝐴-stability and dominating pairs

“…Lemma 3. i/i" 0(z) >0 is valid for every z < 0 and \pn 0 « monotonously descending in the above-mentioned interval. Bearing in mind the result of [1], namely that Rn + k ", k = 0, 1,2, are ^-acceptable, and applying Theorem 4, we readily see that Ehle in [2] and N^rsett in [6] proved that if 0 < 0, then Rfy and äJ£>_, arê -acceptable. Moreover, by applying Theorem 4, we see that for every 0 < 0, u(0) > 0.…”

“…This lemma is valid because, according to [2], R^+k ", 0 < k, I < 1, is A -stable for every n > 1. If we assume that the Ehle conjecture is valid, namely that {Rn+k ", 0 < k < 2, 0 < n} is the set of all A -stable Padé approximations, then Lemma 11 characterizes completely all the dominant pairs which are composed of the approximations Rnl>m. 5.…”

Generalized order star theory

“…Following [12], we call approximations of this form N-approximations. We say that R is a generalized Pad~ approximation to exp(x) [5] if R(x) -exp(x) =: o(Jxl n+m+l-k) and negative distinct x I .... ,x k exist such that R(xi) = exp(xi), i = l,...,k. By [3] and Theorem 4 the only A-acceptable approximations of this kind are obtained for n = m, 0 ~ k ~ 2; for n = m +i, O ~ k ~ 1 and for n = m +2, k = O.…”

Second derivative methods applied to implicit first‐ and second‐order systems

“…We can write this equation as a first-order system Now applying a second derivative formula to (21), from (9) and (10) we obtain g, + h2coM-'F, + h b , M 1 F , + h2co(M-'F:-M-'CM-'F,) where the vectors g1 and g, contain information from previous steps. Our analysis now follows directly from the previous section.…”