Abstract
In this work, we develop a new class of methods which have been created in order to numerically solve non-linear second-order in time problems in an efficient way. These methods are of the Rosenbrock type, and they 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 also follow the ideas of Runge–Kutta–Nyström methods when solving second-order in time problems, we have called them Rosenbrock–Nyström methods. When solving non-linear problems, Rosenbrock–Nyström methods present less computational cost than implicit Runge–Kutta–Nyström ones, as the non-linear systems which arise at every intermediate stage when Runge–Kutta–Nyström methods are used are replaced with sequences of linear ones.
Highlights
IntroductionIt is usual to use numerical methods that have been designed to numerically solve first-order in time problems by transforming the initial problem into a first-order one
In order to avoid all the previous drawbacks when solving a non-linear second-order in time problem, in this paper, we present a new class of methods, which we call Rosenbrock–Nyström methods
As we have seen in this work, these new methods are a good choice when we have a second-order in time problem to solve
Summary
It is usual to use numerical methods that have been designed to numerically solve first-order in time problems by transforming the initial problem into a first-order one In this way, we have several options to choose from, such as Runge–Kutta methods (RK methods), fractional step Runge–Kutta methods (FSRK methods) or exponential integrators. In order to avoid the high computational cost that implicit RKN methods present when multidimensional problems in space are solved, FSRKN methods were developed and studied in [9]. The idea of these methods is to split the spatial operator in a suitable way so that at every intermediate stage the problem to be solved is simpler in a certain way than the original one.
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