Fourth-Order Symmetric DIRK Methods for Periodic Stiff Problems

New symmetric DIRK methods specially adapted to the numerical integration of first-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion conditions for symmetric DIRK methods as well as symmetric stability functions with real poles and maximal dispersion order. Two new fourth-order symmetric methods with four and five stages are obtained. One of the methods is fourth-order dispersive whereas the other method is symplectic and sixth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with the symplectic DIRK method derived by Sanz-Serna and Abia (SIAM J. Numer. Anal. 28 (1991) 1081–1096).