A new technique for the early detection of stiffness in coupled differential equations and application to standard Runge-Kutta algorithms

Abstract.Higher-order Runge-Kutta (RK) algorithms employing local truncation error (LTE) estimates have had very limited success in solving stiff differential equations. These LTEs do not recognize stiffness until the region of instability has been crossed after which no correction is possible. A new technique has been designed, using the local stiffness function (LSF), which can detect stiffness very early before instability occurs. The LSF is a normalized dimensionless ratio which is essentially based on the product of the step size and the geometric mean of all the slopes. It is exceedingly sensitive to the onset of stiffness. Together, the LSF and the LTE form a complementary pair which can cooperate to help solve some mildly stiff equations which were previously intractable to RK algorithms alone. Examples are given of implementation and LSF performance.