Numerical Integration of Systems of Stiff Ordinary Differential Equations with Special Structure

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Abstract
Algorithms for the solution of general systems of stiff differential equations commonly use implicit integration formulae. The associated non-linear equations at each step of the integration are efficiently solved by an iteration such as the parallel chord method, where the matrix is an approximation for the Jacobian at a calculated point of the solution. This iteration frequently gives sufficiently rapid convergence over a number of integration steps before updating and re-inversion of the matrix is required. When the differential equations have a special structure, satisfactory convergence may be maintained by updating a partition of the Jacobian less frequently than the remainder and an efficient computational procedure consists in calculating the corresponding update of the inverse. Sufficient conditions for local convergence may be expressed in terms of the difference between the iteration matrix and the derivative at the solution or in terms of the difference of the corresponding inverses. Similarly the asymptotic rate of convergence is estimated in terms of the norms of these perturbations. To assess the effectiveness of updating a partition of the Jacobian or its inverse we set the corresponding perturbation to zero and evaluate the estimate of the rate of convergence. Variable transformation and “weighting” of equations may be used to obtain more accurate computable estimates of convergence rates and tighter conditions for convergence. This is particularly relevant in non-linear stiff systems arising in applications from physics, chemistry and engineering and associated with fast and slow motions. Such systems exhibit special structure in the Jacobian and higher derivatives related to the sensitivity of the system to the components of the fast motion which makes them particularly amenable to matrix updating techniques. A number of illustrative problems from the literature are cited. A worked example has been solved numerically using an inverse whose partitions are updated irregularly as required for convergence and a comparison is made of iteration counts and inversion statistics with updating of the full inverse. Computational savings may sometimes belarge.

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