The NUMOL solution of time-dependent PDEs using DESI Runge—Kutta formulae Original Research Article

In a recent publication Butcher and Cash introduced a new class of L-stable implicit Runge—Kutta formulae in an attempt to eliminate the disadvantages of SIRK and DIRK formulae and to combine their advantages. These formulae, namely Diagonally Extended Singly Implicit formulae (DESI), have been investigated by the present author and have been implemented in an experimental code in a variable-stepsize/variable-order mode. Numerical experimentation on a large number of stiff systems has shown that the DESI code is much more efficient than STRIDE (which is an implementation of SIRK formulae) and competitive with BDF implementations. In this paper we investigate the efficiency of the DESI code on large dimension stiff systems with banded Jacobians, which arise from the discretization of PDEs with the Numerical Method of Lines (NUMOL). The results obtained by the DESI code are compared with the results obtained by the widely used BDF code LSODE. These results indicate that the current implementation of DESI performs satisfactorily and, in particular, is competitive with LSODE.