Convergence and Accuracy Properties of the Method of Quasilinearization

Abstract

The paper deals with the method of quasilinearization as applied to the solution of two-point boundary-value (TPBV) problems associated with a system ${\bf y}' = {\bf f}(t,{\bf y})$ of second-order differential equations. It is supposed that the equations, linearized about an approximate solution ${\bf y}^{(k)} (t)$, are integrated numerically and that the linear TPBV problem is solved by a scheme such as the method of complementary functions. The principal concern is the effect that the approximate interpolation, used to represent ${\bf y}^{(k)} (t)$, has upon the order of convergence and the order of accuracy of the final converged solution. For example, it is shown that, if Runge-Kutta (RK) integration is employed, then there is an interpolation formula which gives second-order convergence. It is also proven that, if the order of the integration is q (suppose q is even), then the order of the interpolation need be only ${{(q - 2)} / 2}$ to give $(q - 1)$th order global accuracy in the converged solutio...