Some Characteristics of Implicit Multistep Multi-Derivative Integration Formulas

Sufficient conditions for the convergence of methods for the numerical integration of a system of ordinary differential equations, possibly stiff, are presented. The asymptotic error formula for such a solution is developed. The existence of more stringent conditions for convergence of multi-derivative formulas over first derivative formulas is noted. A class of computationally A-stable formulas is presented, with notes on implementing a set of them in a variable stepsize, variable order method. Tests on one such implementation show the capabilities of such methods on stiff problems.