Solution of Differential—Algebraic Systems Using Diagonally Implicit Runge—Kutta Methods

Abstract

Diagonally Implicit Runge—Kutta (DIRK) methods are developed and applied to differential—algebraic systems arising from dynamic process simulation. In particular, an embedded family of DIRK methods is developed for implementation as a variable-step variable-order algorithm. The methods developed allow easy assessment of local solution error as well as the ability to change the order of approximation. The stability properties of the methods are chosen to make them suitable for use on stiff systems. Some important aspects of implementation of DIRK methods are discussed within the context of the solution of differential—algebraic systems. The performance of this algorithm is compared with an alternative variable-order approach based on “triples” which allows the patching together of several fixed-order formulae. The results indicate that the fully embedded DIRK algorithm is generally more efficient than the algorithm based on “triples”. Areas of further investigation in the context of differential—algebraic systems are outlined.