Rosenbrock type methods for solving non-linear second-order in time problems

Abstract

In this work we present a new class of methods which have been developed in order to numerically solve non-linear second-order in time problems. These methods are of Rosenbrock type, and can be seen as a generalization of these methods when they are applied to second-order in time problems which have been previously transformed into first-order in time problems. As they follow the ideas of Runge-Kutta-Nystr¨om methods when solving second-order in time problems, we will call them Rosenbrock-Nystr¨om methods. These new methods present less computational cost than implicit RungeKutta-Nystr¨om ones, as the non-linear systems which arises when RungeKutta-Nystr¨om methods are used are replaced with sequences of linear ones. In this article we show the development of Rosenbrock-Nystr¨om methods, as well as the conditions that must be satisfied to get the desired classical order (up to order four) and the main ideas in order to have stability. Besides, we will show some numerical experiments that prove the good behaviour of these new methods.