We introduce a family of fourth-order two-step methods that preserve the energy function of canonical polynomial Hamiltonian systems. As is the case with linear mutistep and one-leg methods, a prerogative of the new formulae is that the associated nonlinear systems to be solved at each step of the integration procedure have the very same dimension of the underlying continuous problem. The key t...

To integrate large systems of ordinary differential equations (ODEs) with disparate timescales, we present a multirate method with error control that is based on embedded, explicit Runge-Kutta (RK) formulas. The order of accuracy of such methods depends on interpolating certain solution components with a polynomial of sufficiently high degree. By analyzing the method applied to a simple test eq...

In this paper, we present some variants of Cauchy’s method for solving non-linear equations. Analysis of convergence shows that the methods have fourth-order convergence. Per iteration the new methods cost almost the same as Cauchy’s method. Numerical results show that the methods can compete with Cauchy’s method. © 2007 Published by Elsevier B.V. MSC: 41A25; 65D99

Based on a high order absolutely stable implicit scheme, we construct the alternating group explicit iterative method (AGEI) for four order parabolic equations. The method is verified to be convergent, and has the intrinsic parallelism. Results of the convergence analysis shows that the method is of high accuracy.

We reconsider the cosmic string perturbative solution to the classical fourthorder gravity field equations, obtained in Ref. [1], and we obtain that static, cylindricaly symmetric gauge cosmic strings, with constant energy density, can contain only β-terms in the first order corrections to the interior gravitational field, while the exact exterior solution is a conical spacetime with deficit an...