Implicit five-step block method with generalised equidistant points for solving fourth order linear and non-linear initial value problems

Abstract

Abstract Block methods for the solution of fourth order ordinary differential equations are conventionally developed with specific grid or off-grid points. An implicit five-step block method with generalised equidistant points for solving fourth order initial value problems (IVPs) is presented in this article. The generalised points allow for flexibility in choice of points and the strategy adopted for the developing the block method considers the general block form with integration of Taylor series expansion. This results in a family of schemes implemented as simultaneous integrators of the fourth order IVPs at all grid points. The basic properties of the block method are investigated and the block method is seen to satisfy the property of convergence which is evident in the numerical solutions. In addition, the numerical results displayed the five-step block method performing better than existing methods. The block method is also applied to a ship dynamics fourth order model.