Parallel diagonally implicit multistage integration methods for stiff ordinary differential equations

Abstract

From this large family of general linear methods, we look at a special class of methods known as the diagonally implicit multi-stage integration methods (DIMSIMs). One of the four matrices characterizing a DIMSIM, the matrix A, corresponds to the coefficient matrix in a Runge-Kutta method. Hence, the structure of A plays a central role in the implementation costs of the method. We will refer to various choices for the form of A as “types”. In particular, for type 4 methods, A = λI and these are suitable for the parallel solution of stiff problems. A-stable type 4 methods in which the order of the method equals the number of stages, has been derived by J. C. Butcher (1996). In this thesis we consider a variable stepsize, variable order implementation of these methods as parallel solvers for stiff initial value problems. Although these methods have shown potential for the solution of stiff initial value problems they are not very efficient when compared to existing sequential solvers RADAU5 and VODE. This is probably because of the large error constants of these methods. In order to have an efficient set of parallel methods, we derive a new set of A-stable type 4 methods which have one more stage than the order of the method. For these methods we can make the error constants very small and error estimates for stepsize control are available within each step. Hence, these methods are much simpler to implement and in a parallel implementation these methods will cost no more than the methods in which the number of stages is equal to the order. These methods have been successfully implemented in a variable stepsize, variable order code and have been used to solve some well-known stiff problems. On a test problem of dimension 400 these methods have shown speedup factors of up to 2.5. Although the present implementation is slightly slower than the sequential codes such as RADAU5 and VODE, they have shown potential as parallel solvers of stiff initial value problems.