Embedded dirk methods for the numerical integration of stiff systems of odes

Abstract

In this paper, the optimal order of embedded pairs of Diagonally Implicit Runge-Kutta (DIRK) methods is examined. It is shown that a q-stage DIRK method of order p embedded in a q + 1 stage DIRK method of order p + 1 cannot have p = q + 1. Thus adopting embedding techniques to estimate the local truncation error results in giving up an order of accuracy for q<6. Embedded pairs of orders two and three for the basic method are derived with the additional stage being either explicit or implicit. Numerical results indicate that significant savings are realized when the extra stage is explicit.

An analysis of A and L-stability properties of q-stage order q DIRK methods with unequal diagonal elements is presented. Necessary and sufficient conditions for A and L-stability are derived.

To assess the potential of such methods, a number of embedded DIRK formulas are implemented. Numerical results for selected test problems are presented.

Keywords:

• Stiff systems of ordinary differential equations
• Diagonally Implicit Runge-Kutta methods
• A-stability

C.R. Categories:

• G.1.7. Ordinary differential equations
• stiff equations