Hybrid formulas with non-vanishing interval of periodicity

Abstract

Dahlquist [2] established that unconditionally stable linear multistep methods for the solution of the differential equation y′ = f(x,y) is at best of order two and has to be implicit. Lambert and Watson [10] proposed symmetric linear multistep methods of arbitrary step-number which are orbitally stable. Four symmetric hybrid methods namely:

and

with non-vanishing interval of periodicity are hereby proposed for this class of problems which are known to be highly oscillatory. Numerical comparisons are made with some of the existing methods and the number of function evaluations is kept to barest minimum by constraining m≦2. The schemes accommodate adoption of reasonable large mesh-size and facilities stepsize adjustment. Besides they are orbitally stable and compare favourably with the existing schemes.

Keywords:

• Second-order ordinary differential equations
• P-stability
• unconditional stability
• interval of periodicity

AMS (MOS) Subject classification (1970):

• Primary 65L05
• Secondary 65D30

†Research work partially supported by grants from University of Benin, Benin City, Nigeria.

†Research work partially supported by grants from University of Benin, Benin City, Nigeria.

Notes

†Research work partially supported by grants from University of Benin, Benin City, Nigeria.