Maximal order for second derivative general linear methods with Runge-Kutta stability
Applied Numerical Mathematics
A note on multi-step difference schemes
Journal of Computational and Applied Mathematics
Applied Numerical Mathematics
On the construction of second derivative diagonally implicit multistage integration methods for ODEs
Applied Numerical Mathematics
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Given a sequence of integers [n"0,n"1,...,n"r], where n"0,n"r=0 and n"i=-1,i=1,2,...,r-1, a sequence of r polynomials (P"0,P"1,...,P"r) is a generalized Pade approximation to the exponential function if @?"i"="0^rexp((r-i)z)P"i(z)=O(z^p^+^1), where the order of the approximation p is given by p=@?"i"="0^r(n"i+1)-1. The main result of this paper is that if 2n"0p+2, then @?"i"="0^rw^r^-^iP"i(z) is not the stability polynomial of an A-stable numerical method. This result, known as the Butcher-Chipman conjecture, generalizes the corresponding result for rational Pade approximations. The special case, formerly known as the Ehle conjecture [B.L. Ehle, A-stable methods and Pade approximations to the exponential, SIAM J. Math. Anal. 4 (1973) 671-680], was subsequently proved by Hairer, Norsett and Wanner [G. Wanner, E. Hairer, S.P. Norsett, Order stars and stability theorems, BIT 18 (1978) 475-489].