Explicit Runge–Kutta–Nyström methods for the numerical solution of second order linear inhomogeneous IVPs

Abstract

Runge–Kutta–Nyström (RKN) methods for the numerical solution of inhomogeneous linear initial value problems with constant coefficients are considered.A general procedure to construct explicit s-stage RKN methods with maximal order p=s+1, similar to the developed by the authors (Montijano et al., 2023) for the class of second order IVP under consideration, depending on the nodes ci,i=1,…,s is presented. This procedure requires only the solution of successive linear equations in the elements aij, 1≤j<i≤s of the matrix of coefficients A of the RKN method and avoids the solution of non linear equations.The remarkable fact is that using as free parameters the nodes ci,i=1,…,s with a quadrature relation, the s(s−1)/2 elements of matrix A can be computed by solving successively linear systems with coefficients depending on the nodes, so that if they are non-singular we get a unique s-stage method with maximal order s+1.We obtain an optimized six-stage seventh-order RKN method in the sense that the nodes are chosen so that minimize the leading term of the local error. Finally, some numerical experiments are presented to test the behaviour of the optimized RKN method with others with Radau and Lobatto nodes.