An efficient method for solving implicit and explicit stiff differential equations

When the s-stage fully implicit Runge}Kutta (RK) method is used to solve a system of n ordinary di!erential equations (ODE) the resulting algebraic system has a dimension ns. Its solution by Gauss elimination is expensive and requires 2s3n3/3 operations. In this paper we present an e$cient algorithm, which di!ers from the traditional RK method. The formal procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. Its solution is s2/2 times faster than the original, nondiagonalized system, for s even, and s3/(s!1) for s odd in terms of number of multiplications, as well as s2 times in terms of number of additions/multiplications. In particular, for s"3 the method has the same precision and stability properties as the well-known RK-based RadauIIA quadrature of Ehle, implemented by Hairer and Wanner in RADAU5 algorithm. Unlike RADAU5, however, the method is applicable with any s and not only to the explicit ODEs My@"f (x, y), where M"const., but also to the general implicit ODEs of the form f (x, y, y@)"0. The block-diagonal form of the algebraic system allows parallel processing. The algorithm formally di!ers from the implicit RK methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coe$cients are found by enforcing y to satisfy the di!erential equation at the collocation points. Locations of those points are found from the derived stability function such as to guarantee either A- or ¸-stability properties as well as a superior precision of the algorithm. If constructed such as to be ¸-stable the method is a good candidate for solving di!erential-algebraic equations (DAEs). Although not limited to any speci"c "eld, the application of the method is illustrated by its implementation in the multibody dynamics described by both ODEs and DAEs