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High-order P-stable multistep methods. (English) Zbl 0726.65091

A class of two-step methods for solving the initial value problem ${y}^{\text{'}\text{'}}=f\left(t,y\right)$ 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 O. Nevanlinna [SIAM J. Numer. Anal. 20, 1210-1218 (1983; Zbl 0532.65065)]. Hence the schemes are P- stable, of order 2m and implicit.

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 ${f}_{y}$ of degree m. The powers of the Jacobian ${f}_{y}$ 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 I. Gladwell, 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.

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