DESI methods for stiff initial-value problems

Recently, the so-called DESI (diagonally extended singly implicit) Runge-Kutta methods were introduced to overcome some of the limitations of singly implicit methods. Preliminary experiments have shown that these methods are usually more efficient than the standard singly implicit Runge-Kutta (SIRK) methods and, in many cases, are competitive with backward differentiation formulae (BDF). This article presents an algorithm for determining the full coefficient matrix from the stability function, which is already chosen to make the method A-stable. Because of their unconventional nature, DESI methods have to be implemented in a special way. In particular, the effectiveness of these methods depends heavily on how starting values are chosen for the stage iterations. These and other implementation questions are discussed in detail, and the design choices we have made form the basis of an experimental code for the solution of stiff problems by DESI methods. We present here a small subset of the numerical results obtained with our code. Many of these results are quite satisfactory and suggest that DESI methods have a useful role in the solution of this type of problem.