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Zbl 0726.65091
Franco, J.M.; Palacios, M.
High-order P-stable multistep methods. (English)
[J] J. Comput. Appl. Math. 30, No.1, 1-10 (1990). ISSN 0377-0427

A class of two-step methods for solving the initial value problem $y''=f(t,y)$ is presented. The stability polynomial is directly related to the (m,m)-diagonal Padé approximation to the exponential through the $\Sigma$-map, see the reviewer and {\it O. Nevanlinna} [SIAM J. Numer. Anal. 20, 1210-1218 (1983; Zbl 0532.65065)]. Hence the schemes are P- stable, of order 2m and implicit. \par If the implicit equations are solved by a Newton like algorithm at each iteration step the linear system consists of a matrix polynomial in the Jacobian $f\sb y$ of degree m. The powers of the Jacobian $f\sb y$ are avoided by factoring this polynomial. For the linear test equation with an oscillatory forcing term it is shown that the schemes are always in phase in the sense of {\it I. Gladwell}, {\it R. M. Thomas} [Int. J. Numer. Methods Eng. 19, 495-503 (1983; Zbl 0513.65053)]. A numerical comparison with other schemes is presented for linear and nonlinear equations.
[R.Jeltsch (Zürich)]

MSC 2000:: *65L06 Multistep, Runge-Kutta, and extrapolation methods
65L05 Initial value problems for ODE (numerical methods)
65L20 Stability of numerical methods for ODE
34A34 Nonlinear ODE and systems, general

Keywords: P-stability; diagonal Padé approximations to the exponential; two-step methods; stability polynomial; numerical comparison

Citations: Zbl 0532.65065; Zbl 0513.65053

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