Two-step hybrid numerical integrators for the solutions of highly oscillatory systems of ODEs with fixed step size

Collocation methods have shown great promise in developing numerical techniques for solving stiff and highly oscillatory problems in ordinary differential equations. This paper presents the development of a class of block hybrid collocation methods that can effectively solve differential equation systems in block solution form. The block hybrid collocation approaches are based on the highly successful collocation at polynomial nodes. The continuous formulation gives rise to block solution methods, which are discussed for several examples and applications. The techniques generate dense output at both grid and off-grid points within the integration interval and are self-starting. Theoretically, the convergence of the derived methods is established, and asymptotic error constants are computed. For a wide class of problems with oscillatory solutions, better performance is obtained compared to some known standard methods. Numerical experiment using the new methods clearly show improved efficiency and performance compared to several methods with strong algebraic stability properties. The efficiency curves of the solutions are plotted to show how rapidly the hybrid collocation techniques for the second derivative block hybrid methods converge.