In Mitsui (Sci Eng Rev Doshisha Univ Jpn 51(3): [181][182][183][184][185][186][187][188][189][190] 2010) the author introduced a new class of discrete variable methods known as look-ahead linear multistep methods (LALMMs) which consist of a pair of predictor-corrector (PC), including a function value at one more step beyond the present step for the numerical solution of ordinary differential equations. Two-step family of Look-Ahead linear multistep methods of fourth-order pair were derived and shown to be A(θ )-stable in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181-188, 2011). The derived integration methods are of low orders and unfortunately cannot cope with stiff systems of ordinary differential equations. In this paper we extend the concept adopted in Mitsui and Yakubu (Sci Eng Rev Doshisha Univ Jpn 52(3):181-188, 2011) to construct second-derivative of high-order accuracy methods with off-step points which behave essentially like one-step methods. The resulting integration methods are A-stable, convergent, with large regions of absolute stability, suitable for stiff systems of ordinary differential equations. Numerical comparisons of the new methods have been made and enormous gains in efficiency are achieved.

No abstract

Abstract.We consider the construction of an interpolant for use with Lobatto-Runge-Kutta collocation methods. The main aim is to derive single symmetric continuous solution (interpolant) for uniform accuracy at the step points as well as at the off-step points whose uniform order six everywhere in the interval of consideration. We evaluate the continuous scheme at different off-step points to obtain multi-hybrid schemes which if desired can be solved simultaneously for dense approximations. The multi-hybrid schemes obtained were converted to Lobatto-RungeKutta collocation methods for accurate solution of initial value problems. The unique feature of the paper is the idea of using all the set of off-step collocation points as additional interpolation points while symmetry is retained naturally by integration identities as equal areas under the various segments of the solution graph over the interval of consideration. We show two possible ways of implementing the interpolant to achieve the aim and compare them on some numerical examples.Mathematical subject classification: 65L05.

The second derivative block hybrid method for the continuous integration of differential systems within the interval of integration was derived. The second derivative block hybrid method maintained the stability properties of the Runge–Kutta methods suitable for solving stiff differential systems. The lack of such stability properties makes the continuous solution not reliable, especially in solving large stiff differential systems. We derive these methods by using one intermediate off-grid point in between the familiar grid points for continuous solution within the interval of integration. The new family had a high accuracy, non-overlapping piecewise continuous solution with very low error constants and converged under the suitable conditions of stability and consistency. The results of computational experiments are presented to demonstrate the efficiency and usefulness of the methods, which also indicate that the block hybrid methods are competitive with some strong stability stiff integrators.

No abstract