Linear multistep methods for functional-differential equations

Abstract

A new way to define linear multistep methods for functional differential equations is presented, and some of their properties are analyzed. The asymptotic behavior of the global discretization error is investigated. Finally, Milne's device is generalized to functional differen- tial equations. The effect of the nonsmoothness of the exact solution is taken into account. 1. Introduction. Consider the functional differential equation (FDE) (1) x'(t) = F(t, xt) (to < t < T), xto = O, where x(t) E RI, 0 E C((-T,0), Rn), FE C((to,T) x C((-T,0), Rn), RI) and xt: 9 -* x(t + 9) for 0 E ( - T, 0), t E (to, T). It is well known that the solution x of (1) is usually not smooth, that is, even if F and 4 are C? functions, x may have jump discontinuities in its derivatives. The occurrence of these jump discontinuities may lead to order-breakdown for numerical methods if no special provisions are made. It seems that all available techniques require information about the location of the jumps. If the delay is not state-dependent, then these locations may be known a priori. If not, then they may be calculated numerically (see Feldstein and Neves (6)). When the location of a jump discontinuity is known, two different techniques are available: (i) Make the location of the jump a grid point, and restart. In practice, this