Implementation of Falkner method for problems of the form y″= f( x, y)

Abstract

Falkner method (also called Adams extrapolation formulae for second order differential equations) is a multistep code for the numerical integration of second order differential equations. The explicit Falkner method of n steps has order n+1 when calculating the solution and order n when calculating the derivative of the solution. For the special case in which the function does not depend on the derivative of the solution it is possible to increase the order of the computation of the derivative by introducing the implicit formula without increasing the number of evaluations.