Optimal time order when implicit Runge–Kutta–Nyström methods solve linear partial differential equations

We study the order of consistency and convergence which arises when implicit Runge–Kutta–Nyström methods are used for the time discretization of linear second-order in time partial differential equations. The order reduction observed in practice, including its fractional part, is obtained. Using an abstract formulation, it is also proved that this phenomenon can be completely avoided with suitable boundary values for the internal stages. Thanks to recent results of stability, we obtain convergence with optimal order even when arbitrarily stiff space discretizations are integrated by using Runge–Kutta–Nyström methods with infinite interval of stability. The numerical experiments confirm that the optimal order can be reached.