Numerical resolution of linear evolution multidimensional problems of second order in time (Q2885164)

The authors consider an initial value problem of a linear evolution equation of second order in time, which includes a general differential operator defined on a Hilbert space. This abstract operator may contain the spatial derivatives of a corresponding partial differential equation. The authors assume a splitting of the operator into a sum of self-adjoint and negative semi-definite operators. These assumptions are satisfied by the Laplace operator, for example.NEWLINENEWLINENow the authors construct a numerical technique to solve the initial value problem. On the one hand, fractional step Runge-Kutta methods have been applied to partial differential equations with the assumed operator splitting in previous works, see, for example, \textit{B. Bujunda} and \textit{J. C. Jorge} [Appl. Numer. Math. 45, No. 2--3, 99--122 (2003; Zbl 1028.65059)]. On the other hand, Runge-Kutta-Nyström methods have been used to resolve partial differential equations of second order in time, see, for example, \textit{I. Alonso-Mallo, B. Cano} and \textit{M. J. Moreta} [Appl. Numer. Math. 58, No. 5, 539--562 (2008; Zbl 1141.65068)]. Now the authors combine these techniques to obtain the class of fractional step Runge-Kutta-Nyström methods. A relatively low computational effort is achieved by a specific diagonal-implicit structure of the involved coefficients, which guarantees corresponding systems of linear algebraic equations with a relatively small size. The authors prove the consistency of this class of one-step methods under certain assumptions. Moreover, a stability property is defined based on a generalisation of Dahlquist's test equation to the underlying operator splitting. It follows that the approximations obtained by the methods are bounded uniformly for all time step sizes provided that this stability criterion is satisfied.NEWLINENEWLINEFinally, the authors present an example of a fractional step Runge-Kutta-Nyström method, which is consistent of third order due to the theoretical investigations. An artificial partial differential equation is resolved, where a sum of two space derivatives appears. The numerical results confirm the order of convergence for the applied technique.