Iteration of Runge‐Kutta Methods with Block Triangular Jacobians

Abstract

AbstractWe shall consider iteration processes for solving the implicit relations associated with implicit Runge‐Kutta (RK) methods applied to stiff initial value problems (IVPs). The conventional approach for solving the RK equations uses Newton iteration employing the full righthand side Jacobian. For IVPs of large dimension, this approach is not attractive because of the high costs involved in the LU‐decomposition of the Jacobian 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‐diagonal matrix the blocks of which have the IVP dimension. In this paper we study an alternative approach which directly replaces the RK Jacobian by a block‐diagonal or block‐triangular matrix the blocks of which themselves are block‐triangular matrices. Such a grossly ‘simplified’ Newton iteration process allows for a considerable amount of parallelism. However, the important issue is whether this block‐triangular approach does converge. It is the aim of this paper to get insight into the effect on the convergence of block‐triangular Jacobian approximations.