Adaptive step-size approach for Simpson's-type block methods with time efficiency and order stars

In the present scientific literature, block methods to solve stiff and nonlinear initial value problems are in great use due to their better stability features and smaller computational cost. Adaptive step-size versions of such methods, however, are not presented in many research articles, although they are more efficient than their fixed step-size counterparts. Keeping in view the computational efficiency and accuracy obtained with adaptive step-size approaches, two three-step Simpson's-type block methods based on the second derivative having sixth and eighth order of convergence are considered here. To prove their better performance, the results obtained from different numerical simulations with stiff differential systems under the adaptive step-size approach are compared to the results obtained using fixed step-size. Those problems include the Kaps problem, a Gear's problem and the Blasius model from fluid dynamics. When compared to an adaptive step-size version of the well-known Lobatto-IIIA methods (implicit in nature), the superiority of the considered block methods is revealed. To fill the gap in previous research works on the block methods, the theory of order stars is also included herein.