A-stable, Accurate Averaging of Multistep Methods for Stiff Differential Equations

Several low-order numerical solutions of stiff systems of ordinary differential equations are computed by repeated integration, using a multistep formula with parameters. By forming suitable linear combinations of such solutions, higher-order solutions are obtained. If the parameters are properly chosen the underlying solutions, and thus the higher-order one, can be made A-stable and strongly damping with respect to the stiff components of the system. A detailed description is given of an algorithmic implementation of the method, which is computationally efficient. Numerical experiments are carried out on some test problems, confirming the validity of the method.