Blended Implicit Methods for solving ODE and DAE problems, and their extension for second order problems

The use of implicit numerical methods is mandatory when solving general stiff ODE/DAE problems. Their use, in turn, requires the solution of a corresponding discrete problem, which is one of the main concerns in the actual implementation of the methods. In this respect, blended implicit methods provide a general framework for the efficient solution of the discrete problems generated by block implicit methods. In this paper, we review the main facts concerning blended implicit methods for the numerical solution of ODE and DAE problems. In addition to this, we study the extension of blended implicit methods for solving second-order problems, which results in a straightforward generalization of the basic theory for such methods. Finally, a few numerical tests obtained with the computational code BiMD, implementing a variable order-variable stepsize blended implicit method, are also reported, in order to confirm the effectiveness of the approach