REPEATED EXTRAPOLATION TO THE LIMIT IN THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS.
CALIFORNIA UNIV LOS ANGELES
Pagination or Media Count:
Let the initial-value problem y fx,y, x epsilon a,b , ya s, for a system of differential equations be solved numerically by a one-step method or by a linear multistep method see Henrici, Discrete Variable Methods in Ordinary Differential Equations, Wiley, 1962. There results a family of sequences y sub n h which approximate the exact solution y x at the mesh points x x sub n a nh. The parameter h plays an essential role in the construction of the y sub n h. Under suitable regularity conditions on f it is proved that y sub n h has an asymptotic expansion in powers of h, valid as h approaches 0 and n approaches infinity in such a manner that x a nh remains fixed in a,b . These results provide a rigorous justification for the algorithm of repeated extrapolation to the limit in which successive terms of the expansion for the error y sub n h - yx are removed by making use of the knowledge of the approximate solutions computed with the values h 2 to the minus m power h sub O, m 0,1, . . . . The convergence properties of the algorithm are discussed in some generality and the theory is verified experimentally for a number of classical numerical methods. Author