ITERATION OF RUNGE-KUTTA METHODS BLOCK TRIANGULAR JACOBIANS

We shall consider iteration processes for solving the implicit relatio ns associated with implicit Runge-Kutta (RK) methods applied to stiff initial value problems (IVPs). The conventional approach for solving t he RK equations uses Newton iteration employing the full righthand sid e Jacobian. For IVPs of large dimension this approach is not attractiv e because of the high costs involved in the LU-decomposition of the Ja cobian of the RK equations. Several proposals have been made to reduce these high costs. The most well-known remedy is the use of similarity transformations by which the RK Jacobian is transformed to a block-di agonal matrix the blocks of which have the IVP dimension. In this pape r we study an alternative approach which directly replaces the RK Jaco bian, by a block-diagonal or block-triangular matrix the blocks of whi ch themselves are block-triangular matrices. Such a grossly 'simplifie d' Newton iteration process allows for a consider able amount of paral lelism. However, the important issue is whether this block-triangular approach does converge. It is the aim of this paper to get insight int o the effect on the convergence of block-triangular Jacobian approxima tions.