Two-stage and Three-stage Diagonally Implicit Runge-Kutta Nystrom Methods of Orders Three and Four

We investigate the existence of two-stage and three-stage R-stable,P-stable, RL-stable, and dispersively enhanced diagonally implicitRunge-Kutta Nyström methods of orders three and four. Wefirst show that a one-parameter family of two-stage third-orderR-stable diagonally implicit methods exists, and that theirdispersive order is at most four. From this we show that two-stagefourth-order R-stable, and third-order P-stable and RL-stablediagonally implicit methods do not exist. Next we show a two-parameterfamily of three-stage fourth-order R-stable diagonally implicitmethods exists with dispersive order at most four, and thatthis family contains a one-parameter family of P-stable methodsand a unique RL-stable. We also show that a one-parameter familyof fourth-order diagonally implicit methods with dispersiveorder at least six exists, and that they are not R-stable. Wepresent third- and fourth-order R-stable and P-stable methodswith small principal truncation coefficients and discuss howthese methods might be implemented in an efficient integrator.