Construction of Two-Derivative Runge–Kutta Methods of Order Six

Two-Derivative Runge–Kutta methods have been proposed by Chan and Tsai in 2010 and order conditions up to the fifth order are given. In this work, for the first time, we derive order conditions for order six. Simplifying assumptions that reduce the number of order conditions are also given. The procedure for constructing sixth-order methods is presented. A specific method is derived in order to illustrate the procedure; this method is of the sixth algebraic order with a reduced phase-lag and amplification error. For numerical comparison, five well-known test problems have been solved using a seventh-order Two-Derivative Runge–Kutta method developed by Chan and Tsai and several Runge–Kutta methods of orders 6 and 8. Diagrams of the maximum absolute error vs. computation time show the efficiency of the new method.