An Alternative Implementation of Variable Step-Size Multistep Formulas for Stiff ODEs

Abstract

An alternative technique for the implementation of variable step-size multistep formulas is developed for the numerical solution of ordinary differential equations. Formulas based on this technique have the property that their leading coefficients are constant; thLs LS important for methods which solve stiff systems. In addition, both theoretmal and empirical results indicate that methods based on this techmque have stability properties similar to those of the corresponding variable coefficient implementations. As a particular example, we have nnplemented the backward differentiation formulas m this form, and the numerical results look very promising. vary the step size and possibly the order to take as large a step as possible consistent with a local control of the estimated error. Two techniques are commonly used to implement variable step-size multistep methods. One technique is based on fixed coefficient formulas and uses interpolation to generate approximations to the solution at evenly spaced past points; DIFSUB [10] and GEAR [14] are two well-known codes based on fixed coefficient formulas. The other technique uses variable coefficient formulas which do not require past values to be evenly spaced; EPISODE [3] and BDF [2] are examples of codes based on variable coefficient formulas. Each of the methods mentioned above Permission to copy without fee all or part of this material LS granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notme and the tltle of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwme, or to republish, requires a fee and/or specific permission.