Parallel-iterated pseudo two-step Runge-Kutta methods with step size control

The aim of this paper is to develop a class of constant step size parallel-iterated pseudo two-step Runge-Kutta methods (PIPTRK methods) for nonstiff first-order ODE problems into variable step size methods. Embedded formulas are provided for giving a cheap error estimate used in the step size control. Methods with variable parameters approach were applied for overcoming the difficulty in using two-step methods with variable step size. By applications to a few widely used test problems, we compare the efficiency of the resulting PIPTRK methods with step size control (PIPTRKSC methods) with the codes PIRK, DOPRI5, DOP853 and ODEX. This numerical comparison shows that these new PIPTRKSC methods are by far superior to the PIRK, DOPRI5, DOP853 and ODEX codes.