An Adaptive Step-Size Optimized Seventh-Order Hybrid Block Method for Integrating Differential Systems Efficiently

Abstract

This article proposes a novel adaptive step-size numerical method for solving initial value ordinary differential systems. The development of the proposed method is based on the theory of interpolation and collocation in which representation of the theoretical solution of the problem is assumed in the form of an appropriate interpolating polynomial. In order to bypass the first Dahlquist’s barrier on linear multistep methods, the proposed method considers five intra-step points in one-step block $\left[ x_n,x_{n+1}\right]$ resulting in a hybrid method. Among these considered five intra-step points, the values of two intra-step points were fixed named as supporting off-step points and the optimized values of the other three intra-step points were obtained by minimizing the local truncation errors of the main formula at the point $x_{n+1}$ and other two additional formulas at supporting off-step points. The proposed method exhibits the property of self-starting as the formulation is immersed into a block structure which enhances the efficiency of the method. The resulting method is of order seven retaining the characteristic of $\mathcal {A}$-stability. The precision of numerical solution is intensified by drafting the proposed algorithm into an adaptive step-size formulation using an embedded-type procedure. The adaptive step-size method has been tested on some well-known stiff differential systems, viz., Robertson’s chemistry problem, Gear’s problem, the Brusselator system, Jacobi elliptic functions system, etc. The proposed method performs well in comparison to other iconic codes available in the literature.