High-order Padé and Singly Diagonally Runge-Kutta schemes for linear ODEs, application to wave propagation problems

In this paper we address the problem of constructing high-order implicit time schemes for wave equations. We consider two classes of one-step A-stable schemes adapted to linear Ordinary Differential Equation (ODE). The first class, which is not dissipative is based upon the diagonal Pade approximant of exponential function. In this class, the obtained schemes have the same stability function as Gauss Runge-Kutta (Gauss RK) schemes. They have the advantage to involve solution of smaller linear system at each time step compared to Gauss RK. The second class of schemes are constructed such that they require the inversion of a unique linear system several times at each time step like the Singly Diagonally Runge-Kutta (SDIRK) schemes. While the first class of schemes is constructed for an arbitrary order of accuracy, the second class schemes is given up to order 12. The performance assessment we provide shows a very good level of accuracy with very reduced computational costs for both class of schemes. But diagonal Pade schemes seem to be more accurate and more robust.