Mehdi Babaei Ghalehjoogh

The main concern of this paper is to develop a new class of A-stable fourth-order numerical scheme for solving initial value problems. The idea is an evolutionary and heuristic approach; using the Grasshopper optimization, along with the Hermite interpolation for stages, we obtain a class of A-stable methods. Four types of weighting rules are introduced for the current formulation. The fundamental weighting rule (FWR) is the most important rule, which emphasizes on the symmetric and central structure of the method. A systematic strategy is proposed to obtain the FWR based on swarm intelligence and regression. The new techniques are called αI-(e + i)P, where e and i are the number of terminal and internal points, respectively. The numerical experiments demonstrate the reasonable behaviour of the algorithms on several test problems from different applications. Finally, we find that the new formulas are well suited for long time behaviour of the time-history analysis in earthquake engineering.