Exploiting the separability in the solution of systems of linear ordinary differential equations

Abstract

Systems of s linear ordinary differential equations (ODEs) are considered. A system of ODEs is separable if the number of stiff eigenvalues of the Jacobian matrix is k « s. A code, DENS2, for solving separable systems of linear ODEs is compared with the corresponding code, DENS1, in which the same integration algorithm is implemented, but the separability is not exploited. A family of test-examples is introduced and used to carry out systematic comparisons between DENS1 and DENS2. The main topics in the investigation are: (i) the efficiency of DENS2 as a function of k/ s, (ii) the performance of DENS2 for very stiff problems, (iii) the behaviour of DENS2 when the solution of the system of ODEs is rapidly varying, (iv) the performance of DENS2 when the Jacobian matrix is slowly varying and (v) the behaviour of DENS2 for different gaps between the stiff and non-stiff eigenvalues. The storage requirements for DENS2 are also discussed. A vectorized version of DENS2 is developed and it is shown that the separability algorithm implemented in DENS2 is even more efficient on vector processors than on sequential computers. The ideas applied in the development of DENS2 can easily be extended (a) for two wide classes of integration algorithms (the backward differentiation formulae and the methods of Runge-Kutta type) and (b) for systems of nonlinear ODEs.