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Superstable two-step methods for the numerical integration of general second order initial value problems. (English) Zbl 0579.65073
The paper presents a class of two-step fourth order methods for the numerical integration of the general second order initial value problem. These methods are implicit and need an iterative process at each step. The author proves that the considered methods are superstable for the test equation $y''+2\alpha y'+\beta\sp 2y=0$, $\alpha,\beta >0$, $\alpha +\beta >0$. Such a kind of equation occurs, for example, in the numerical integration of singular perturbation problems. In my opinion more studies should be done on the practical application of these new methods. First of all it refers to the comparison of the methods proposed by the author with the other well-known methods for solving the general second order initial value problem. In this paper the author does not present any numerical examples.
Reviewer: A.Marciniak

##### MSC:
 65L05 Initial value problems for ODE (numerical methods) 65L20 Stability and convergence of numerical methods for ODE 34A34 Nonlinear ODE and systems, general
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##### References:
 [1] Chawla, M. M.: Unconditionally stable noumerov type methods for second order differential equations. Bit 23, 541-542 (1983) · Zbl 0523.65055 [2] Chawla, M. M.: A fourth-order triadiagonal finite difference method for general non-linear two-point boundary value problems with mixed boundary conditions. J. inst. Math. applics. 21, 83-93 (1978) · Zbl 0385.65038 [3] Dahlquist, G.: On accuracy and unconditional stability of linear multistep methods for second order differential equations. Bit 18, 133-136 (1978) · Zbl 0378.65043 [4] Lambert, J. D.: Stiffness. Computational techniques for ordinary differential equations (1980)