Differential/Algebraic Equations As Stiff Ordinary Differential Equations

Abstract

This paper deals with the relation between differential/algebraic equations (DAEs) and certain stiff ODEs and their respective discretizations by implicit Runge–Kutta methods. For that purpose for any DAE a singular perturbed ODE is constructed such that the DAE is its reduced problem and the solution of the ODE converges in some sense to that of the DAE. Thus the DAE can be interpreted as an infinitely stiff ODE. An analysis of the discretization error of this singular perturbed system gives insight into the relationship of order-reduction phenomena observed for stiff ODEs to that for DAEs. Analysis of a general class of singularly perturbed problems and their discretizations is not attempted; however, the technique of treating singularly perturbed problems and DAEs in a unified way is new and can possibly be applied to other systems and their discretizations as well. Since asymptotic expansions are not used, but an approach similar to the ones used in B-convergence theory is applied, one can derive error bounds that are uniform in the perturbation parameter as well as in the stepsize and do not suffer from restrictions on the ratio of these parameters. This enables one to relate the order of convergence achieved for DAEs to the order of B-convergence. This phenomenon is discussed for several classes of Runge–Kutta methods and illustrated with a numerical example.