Recursive collocation for the numerical solution of stiff ordinary differential equations

Abstract:

The exact solution of a given stiff system of nonlinear (homogeneous) ordinary differential equations on a given interval I is approximated, on each subinterval ${\sigma _k}$ corresponding to a partition ${\pi _N}$ of I, by a linear combination ${U_k}(x)$ of exponential functions. The function ${U_k}(x)$ will involve only the "significant" eigenvalues (in a sense to be made precise) of the approximate Jacobian for ${\sigma _k}$. The unknown vectors in ${U_k}(x)$ are computed recursively by requiring that ${U_k}(x)$ satisfy the given system at certain suitable points in ${\sigma _k}$ (collocation), with the additional condition that the collection of these functions $\{ {U_k}\}$ represent a continuous function on I satisfying the given initial conditions.