An optimized two-step hybrid block method for solving general second order initial-value problems

TL;DR: In this paper, the integration of systems of second-order linear inhomogeneous initial value problems with constant coefficients is considered, and a hybrid Numerov method is used that is constructed in the sense of Runge-Kutta ones.

TL;DR: In this paper, a three-stages two-stage method was proposed to solve the Schrodinger problem in high algebraic order, where the approximation of the first layer is done on the point of the point $$x n-1} and not on the usual point.

TL;DR: In this article, the integration of the special second-order initial value problem is considered and a new family of hybrid Numerov methods attaining eighth algebraic order is given at a cost of only 7 function evaluations per step.

Abstract: We consider the integration of the special second‐order initial value problem of the form y′′=f(x,y) . A recently introduced family of 7 stages, eighth‐order methods, sharing constant coefficients, is used as base. This family is properly modified to derive phase fitted and zero dissipative methods (ie, trigonometric fitted) that are best suited for integrating oscillatory problems. Numerical tests over a set of problems shows enhanced performance when the purely linear part of the problems is rather large in comparison with the rest of nonlinear parts. An appendix implementing a MATLAB listing with the coefficients of the new method is also given.

TL;DR: This paper presents a meta-modelling framework for generalized backward Differentiation Formulae, and some of the methods used in this framework have been adapted for practical use in the reinforcement learning environment.

TL;DR: From the numerical results obtained by the new method to well-known periodic problems, show the superior efficiency, accuracy, stability of the method presented.

TL;DR: In this article, a class of implicit one-step methods for solving ordinary differential equations which generalize the trapezoidal rule is studied and the asymptotic behavior of both implicit and predictor-corrector procedures is examined.

TL;DR: In this article, the Butcher-type results for Nystrom-methods have been derived for the first order differential equations of thesecond order and systems of the second order, and a theory for numerical methods for thesecond-order differential equations has been developed.