An efficient method for solving implicit and explicit stiff differential equations

Abstract

SUMMARYWhen 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 2sn/3 operations. In this paper we present an e$cient algorithm, which di!ers from the traditional RK method. The formal procedure for uncoupling the algebraic system into a block-diagonal matrix with s blocks of size n is derived for any s. Its solution is s/2 times faster than the original, nondiagonalized system, for s even, and s/(s!1) for s odd in terms of number of multiplications, as well as s times in terms of number of additions/multiplications. In particular, for s"3 the method has the same precision and stability properties as the well-known RK-based RadauIIA quadrature of Ehle, implemented by Hairer and Wanner in RADAU5 algorithm. Unlike RADAU5, however, the method is applicable with any s and not only to the explicit ODEs My"f (x, y), where M"const., but also to the general implicit ODEs of the form f (x, y, y)"0. The block-diagonal form of the algebraic system allows parallel processing. The algorithm formally di!ers from the implicit RK methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coe$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.

Diagonally Implicit Block Backward Differentiation Formula with Optimal Stability Properties for Stiff Ordinary Differential Equations

“…Figure 1 illustrates the fully implicit and diagonally implicit classes of Runge-Kutta (RK) methods, which can be defined as a 3 × 3 matrix. As stated in [15], RK methods are characterized by excellent stability properties that make them useful for solving stiff ODEs systems. However, for a Fully Implicit Runge-Kutta (FIRK) method, a system of n × r non-linear equations must be solved in each of its integration stage, where n is the dimension of the problem and r is the number of stages, as described by [16].…”