On the Numerical Solution of Stiff Linear Systems of the Oscillatory Type

Abstract

A method is presented for the numerical solution of a stiff linear system of ODE., $\dot X(t) = P(t)X(t) + F(t),\quad X(0) = X_0 .$ The stiffness is assumed to arise from large imaginary parts of some of the eigenvalues of $P(t)$. The large imaginary parts cause a rapidly oscillating component to be superimposed on $X(t)$ forcing the use of very small integration steps. This difficulty is overcome by “filtering out” the large eigenvalues making it possible to calculate the smooth component of X with a reasonable stepsize. An explicit formula is also given for the oscillatory component of X. Finally, results of numerical experiments are presented. A second article by the same authors is to appear which includes another method of solution for similar kinds of problems.