Adaptive stiff solvers at low accuracy and complexity

This paper is concerned with adaptive stiff solvers at low accuracy and complexity for systems of ordinary differential equations. The considered stiff solvers are: two second order Rosenbrock methods with low complexity, and the BDF method of the same order. For the adaptive algorithm we propose to use a monitor function defined by comparing a measure of the local variability of the solution times the used step size and the order of magnitude of the solution instead of the classical approach based on some local error estimation. This simple step-size selection procedure is implemented in order to control the behavior of the numerical solution. It is easily used to automatically adjust the step size, as the calculation progresses, until user-specified tolerance bounds for the introduced monitor function are fulfilled. This leads to important advantages in accuracy, efficiency and general ease-of-use. At the end of the paper we present two numerical tests which show the performance of the implementation of the stiff solvers, with the proposed adaptive procedure.