A -Stable Composite Multistep Methods

Abstract

Consider the set of multistep formulas ∑ ^{l -1} j _mn - k α _ij x _{mn + j} - h ∑ ^{l -1} j _mn - k β _ij x _{mn + j} = 0, i = 1, ···, l , where x _{mn + j} = y _{mn + j} for j = - k , ···, -1 and x _n = ƒ _n = ƒ( x _n , t _n ). These formulas are solved simultaneously for the x _{mn + j} with j = 0, ···, l - 1 in terms of the x _{mn + j} with j = - k , ··· , - 1, which are assumed to be known. Then y _{mn + j} is defined to be x _{mn + j} for j = 0, ··· , m - 1. For j = m , ··· , l - 1, x _{mn + j} is discarded. The set of y 's generated in this manner for successive values of n provide an approximate solution of the initial value problem: y = ƒ( y, t ), y ( t ₀ ) = y ₀ . It is conjectured that if the method, which is referred to as the composite multistep method, is A -stable, then its maximum order is 2 l . In addition to noting that the conjecture conforms to Dahlquist's bound of 2 for l = 1, the conjecture is verified for k = 1. A third-order A -stable method with m = l = 2 is given as an example, and numerical results established in applying a fourth-order A -stable method with m = 1 and l = 2 are described. A -stable methods with m = l offer the promise of high order and a minimum of function evaluations—evaluation of ƒ( y, t ) at solution points. Furthermore, the prospect that such methods might exist with k = 1—only one past point—means that step-size control can be easily implemented