The Solution of Systems of Stiff Nonlinear Differential Equations by Recursive Collocation Using Exponential Functions

Abstract

A system of stiff nonlinear (homogeneous) ordinary differential equations, Y′ = F(x,Y), xεI: = [a,b], with Y(a) = Y₀ (YεRⁿ, n > 1; F(x,0) = 0, xεI) is solved numerically by approximating the solution Y(x) on each subinterval of a given partition π_N: a = t₀ < t₁ < ... < t_N = b by a function of the form ${U_k}\left( x \right) = \sum\limits_{j = 1}^m {{C_{k.j}}} \exp ({\lambda _{k.j}}(x - {t_k}))\left( {{C_{k.j}}\varepsilon {R^n}} \right)$, where m = m(k) ≧ 1. The quantities {λ_k.j} are given: they are the eigenvalues (assumed to be real) of the (approximate) Jacobian of the system at x = t_k. The function U(x) defined by the collection of the functions {U_k(x)} is to be continuous on I. The vectors {C_k.j} are determined recursively by collocation on a finite subset of I and by observing the continuity requirements at x = t_k, k = 1,...,N−1. In the linear case, f(x,Y) = AY (where A is a constant matrix) the method is equivalent with the method of projecting the exact solution into the subspace spanned by certain of the eigenvectors of A. It is shown that the method of recursive collocation is easily modified for the cases where some of the eigenvalues {λ_k.j} are complex, or where the system under consideration is inhomogeneous.