Stiffness Detection Revisited

Many applied differential equations exhibit some form of stiffness, which restricts the step–size and, hence, effectiveness of explicit solution methods. A number of implicit methods have been proposed to circumvent this problem. However, implicit methods can also be substantially less efficient, due to overhead associated with the intrinsic linear algebra. Several attempts have been made to provide user–friendly codes, that would automatically attempt to detect stiffness, at run time, and switch between appropriate methods as necessary. In this work, we outline a new implementation to automatically equip a code with a stiffness detection device. Particular attention is given to the problem of estimation of the dominant eigenvalue of a matrix. We propose an efficient implementation, based on subspace and Krylov iteration, that is now part of the automated method selection used in the scientific problem solving environment of Mathematica. To demonstrate the effectiveness of our strategy, numerical experiments are given with a focus on stiff differential systems that arise in bio-medical applications.