A stepsize control strategy for stiff systems of ordinary differential equations

In solving stiff systems of ordinary differential equations using BDF methods, Jacobians needed for quasi-Newton iteration are frequently computed using finite differences. Round-off errors in the finite-difference approximation can lead to Newton failures forcing the code to choose its time steps based on “stability” rather than accuracy considerations. When standard stepsize control is used, the code can experience thrashing which increases the total number of time steps, Jacobian evaluations, and function evaluations. In this paper we investigate this situation, explaining some surprising time step selection behavior produced by the standard control mechanism. A new control mechanism is proposed which attempts to find and use a “stability” stepsize. A comparison of the new strategy with the standard strategy and with two PI controllers introduced earlier is made using the stiff test set.