Special Second-order Differential Equations Research Articles

Multi-step hybrid methods for special second-order differential equations y ″(t) = f(t,y(t))

In this paper, multi-step hybrid methods for solving special second-order differential equations yź(t) = f(t,y(t)) are presented and studied. The new methods inherit the frameworks of RKN methods and linear multi-step methods and include two-step hybrid methods proposed by Coleman (IMA J. Numer. Anal. 23, 197---220, 8) as special cases. The order conditions of the methods were derived by using the SN-series defined on the set SNT of SN-trees. Based on the order conditions, we construct two explicit four-step hybrid methods, which are convergent of order six and seven, respectively. Numerical results show that our new methods are more efficient in comparison with the well-known high quality methods proposed in the scientific literature.

Two-step-by-two-step PIRKN-type PC methods based on Gauss–Legendre collocation points for nonstiff IVPs

This paper investigates parallel predictor–corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations of the form \(\mathbf{y}''(t)=\mathbf{f}(t,\mathbf{y}(t))\). The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of \(s\)-stage collocation Gauss–Legendre Runge–Kutta–Nystrom (RKN) methods based on \(\hat{c}_1,\dots ,\hat{c}_s\) and \(2s\)-stage collocation RKN methods based on \(\hat{c}_1,\dots ,\hat{c}_s,1+\hat{c}_1,\dots ,1+\hat{c}_s\). At \(n\)th integration step, the stage values of the \(2s\)-stage collocation RKN methods evaluated at \(t_n+(1+\hat{c}_1)h,\dots ,t_n+(1+\hat{c}_s)h\) can be used as the stage values of the collocation Gauss–Legendre RKN method for \((n+2)\)th integration step. In this way we obtain the corrector methods in which the integration processes can be proceeded two-step by two-step. The resulting parallel PC iteration methods which are called two-step-by-two-step (TBT) parallel-iterated RKN-type (PIRKN-type) PC methods based on Gauss–Legendre collocation points (two-step-by-two-step PIRKNG methods or TBTPIRKNG methods) give us a faster integration process. Fixed stepsize applications of these TBTPIRKNG methods to three widely used test problems show that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RKN methods (PIRKN methods) and sequential codes ODEX2 and DOP853 available from the literature.

Explicit multi-step peer methods for special second-order differential equations

The construction of s-stage explicit two- and three-step peer methods of order p = 2 s and p = 3 s is considered for the solution of non-stiff second-order initial value problems where the right-hand side does not depend on y′. By numerical search with respect to large stability intervals and small error constants sequential and parallel methods are constructed. Numerical tests of these peer methods in MATLAB and comparisons with a Runge–Kutta–Nyström method show the efficiency of the proposed methods.

An improved class of generalized Runge–Kutta–Nyström methods for special second-order differential equations

We propose a new family of explicit methods of order four with two evaluations per step, for the numerical integration of special second-order differential equations given by y ″= f( y). These two-stage formulas can be seen as a generalization of the explicit two-stage Runge–Kutta–Nyström methods, providing better order and stability results. We will show that it is possible to obtain methods that are more efficient than the classical Runge–Kutta–Nyström one-step methods with the same number of evaluations per step, specially when highly oscillatory problems are considered. Some numerical experiments are discussed in order to show the good performance of the new schemes.

Explicit pseudo two-step RKN methods with stepsize control

This paper is devoted to variable stepsize strategy implementations of a class of explicit pseudo two-step Runge–Kutta–Nyström methods of arbitrarily high order for solving nonstiff problems for systems of special second-order differential equations. The constant stepsize explicit pseudo two-step Runge–Kutta–Nyström methods are developed into variable stepsize ones and equipped with embedded formulas giving a cheap error estimate for stepsize control. By two examples of widely-used test problems, a pseudo two-step Runge–Kutta–Nyström method of order 8 implemented with variable stepsize strategy is shown to be much more efficient than parallel and sequential codes available in the literature. With stringent error tolerances, this new explicit pseudo two-step Runge–Kutta–Nyström method is even superior to sequential codes in a sequential computer.